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Class 
Book 



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CflBauGwr BEPOsm 



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Handbook 
of 

Engineering Mathematics 



BY 



WALTER E. WYNNE, B. E. 



AND 



WILLIAM SPRARAGEN, B. E. 



lis ILLUSTRATIONS 




NEW YORK 

D. VAN NOSTRAND COMPANY 

25 Park Place 
1916 






Copyright, 1916, 

BY 

D. VAN NOSTRAND COMPANY 



JAN -2 1917 

Stanbopc jprcss 

F. H. GILSON COMPANY 
BOSTON, U.S.A. 



CI,A453449 



AUTHORS' NOTE 



The authors have to express their thanks to Profes- 
sors Irving P. Church, G. A. Goodenough, and William 
A. Granville, who have kindly given permission for the 
use of special material, tables, and constants from their 
works, and to whom proper credit is given where such 
material appears. Thanks are also due to John D. Ball, 
of the Consulting Engineering Department of the Gen- 
eral Electric Company, for coefficients of hysteresis 
loss in iron. 

The authors are especially indebted to Professors 
Ernst J. Berg and John N. Vedder of Union College, 
and to Professor William D. Ennis of the Brooklyn 
Polytechnic Institute, for a critical reading of the 
manuscript and for valuable suggestions; also to Pro- 
fessors Charles F. F. Garis and Walter L. Upson of 
Union College for advice in connection with certain 
sections. 

August, 1916. 



Ill 



INTRODUCTION 

BY 

ERNST J. BERG, Sc. D. 

Professor of Electrical Engineering in Union College, 

Consulting Engineer of the General Electric 

Company, Schenectady, N. Y, 

In publishing this book the authors have endeavored 
to supply a handy means of reference to theoretical and 
apphed mathematics used in engineering, and while the 
first aim has been to make this a mathematical hand- 
book, the book is of greater value because it includes 
the imderlying engineering data and applications as well 
as the mathematical formulae. 

It is intended primarily for students in engineering 
schools and colleges, and should serve as a convenient 
reminder of things which are easily forgotten but are 
likely to be needed in their later work. 

In including differential equations, the authors have 
gone as far as seems necessary for students and en- 
gineers who have taken the ordinary undergraduate 
college course. 

The increasing need of mathematics in engineering 
should assure this book a broad field of usefulness, not 
only to students in technical schools and colleges but 
also to practising engineers. 

E. J. B. 



TABLE OF CONTENTS 



Page 

Algebra i 

Exponents i 

Binomial theorem 2 

Progressions 2 

Logarithms . , 3 

Series 5 

Complex imaginary quantities 7 

Geometry 10 

Plane figures 10 

Solids 12 

Plane Trigonometry 13 

Niunerical values 14 

Trigonometric formulae 14 

Plane triangles 16 

Spherical Trigonometry 16 

Formulae 16 

Application to navigation 17 

Plane Analytic Geometry 19 

Straight line 19 

Rectangular and polar coordinates 20 

Circle 21 

Parabola 22 

Ellipse 23 

Hyperbola 24 

Cycloid 25 

Epicycloid 26 

Hypocycloid 26 

Catenary 27 

Solid Analytic Geometry 27 

Direction cosines 27 

vii 



VIU TABLE OF CONTENTS 

Page 

Plane 29 

Straight line 30 

Calculus 30 

Application of differential calculus 30 

Formulae of differential calculus 33 

Maxima and minima 34 

Application of integral calculus 36 

Curve tracing 40 

Methods of integration 41 

Table of integrals 46 

Hyperbolic Functions 54 

HyperboUc transformations 54 

Hyperbolic formulae 55 

Inverse hyperbolic functions 56 

Differentials of hyperboHc functions 57 

Differential Equations 58 

Equations of the first order, and first degree 59 

Equations of the first order, higher than the first degree 62 

Linear equations with constant coefficients 63 

Homogeneous linear equations 68 

Exact differential equations 69 

Equations of the second order and first degree 70 

Theoretical Mechanics 72 

Center of gravity 72 

Moment of inertia 75 

Motion of a body 78 

Falling bodies 80 

Projectiles 80 

Impact .' 86 

Forces 87 

Friction 88 

Inclined plane. 89 

Mechanics of Materials 91 

Direct stress , 92 

Strength of materials 93 

Torsion 94 

Flexure of beams 95 



TABLE OF CONTENTS IX 

Page 

Shear 97 

Beam loadings 98 

Columns 106 

Center of gravity iii 

Moment of inertia of solids 115 

Properties of standard sections 116 

Hydraulics 119 

Head and pressure 119 

Center of pressure 119 

Flow through apertures 120 

Flow of water in pipes 121 

Losses in pipes 122 

Flow through open channels 127 

Flow over weirs 129 

Stresses in pipes and cylinders 130 

Flow or Fluids 131 

Flow through apertures 131 

Flow through pipes 132 

Electricity 134 

Resistivity 134 

Temperature coefficient 135 

Copper wire tables 136 

Motors and generators 139 

Induced voltage 142 

Inductance 143 

Capacity 145 

Alternating currents 147 

Magnetism 154 

Measurement 158 

English weights and measures 158 

English and metric conversion factors 160 

Heat, electric, and mechanical equivalents 164 

Pressure equivalents 165 

Pressure and volimie corrections 166 

Physical and Chemical Constants 168 

Atomic weights 168 

Weight and density of various substances 169 



X TABLE OF CONTENTS 

Page 

Melting and boiling points 172 

Specific heats 173 

Coefficients of linear expansion of solids 176 

Properties of saturated steam 177 

Tables 185 

Circumferences and areas of circles 185 

Powers, roots, and reciprocals 187 

Common logarithms 192 

Natural logarithms 194 

Trigonometric functions 197 

Hyperbolic sines and cosines 213 



Engineering Mathematics 

ALGEBRA 
Quadratic Equations 

ax^+ bx + c = X = ^ 

2 a. 

The term b^ — 4: ac, called the discriminant, deter- 
mines the nature of the roots. If 6^ is greater than 
4 ac, the roots are real. If W' is less than 4 ac, the roots 
are imaginary. And if 6^ = 4 ac, the roots are real 

and equal. 

Exponents 

1 1 

a^ = — a~^ = — 

{cry = a'^ri 

Special and Indeterminate Forms 

a« = 1 

^0°= cx), a> \ 

a-°° = -4: = — = 0, a>\ 

- = 00 — = 

00 

00 _ 

— =00 - = 

a a 

• 00, 77, — , 0^ 1°^, 00^, 00—00 are indeterminate. 
' oo' 

For the evaluation of indeterminate forms, see 

page 38. 



2 ENGINEERING MATHEMATICS 

Binomial Theorem 

/yt fyyt 1 J 

{x + yY = x^ + nx^'-^y H ^-y^ — - x^'-'^y^ 

. nin — V) (n— 2) ^ « , , 
+ -^ ^ x'^-^y^ + • • • 

(\+xY=\+nx-{- ^ ' x^+— 3!~^ "*" ' * ' 

Arithmetical Progression 
An arithmetical progression is one whose terms 
increase or decrease by a common difference, 
a, a-\- dj a-\- 2d, a-\r ^ d, . . . 

the last term is L = a-{- {n — 1) d 
the sum of the terms is 

S = '^{a + L) ='^[2a+ in- \) d] 

a = first term 

n = number of terms 

d = common difference 

Geometrical Progression 

Quantities are in geometrical progression when 
each term is equal to the preceding term multipUed by 
a constant, 

a, ar, ar^, ar^, . . . 

the last term is L = ar""'^ 

the sum of the terms is 

_ a (f^ — 1) _ g (1 — r^) _ rL— a 
r — 1 1 — r r— 1 

a = first term 
r = constant ratio 
n = number of terms 



ALGEBRA 3 

The sum of an infinite number of terms in geometrical 
progression is 

S= ^ 



l-r 

in which the ratio r must be less than 1 if the series is 
to be convergent (see Infinite Series) . 

Logarithms 

The logarithm of any number to a given base is the 
power to which the base must be raised in order to 
produce the given number, thus: 

if x"^ = y, then m = logxy, 

that is, m is the logarithm of y to the base x. 
The following relations hold for any base: 

log ab = log a + log b 

log - = log a - log b 
log a^ = n log a 

log- = - loga 

The base of the common system of logarithms is 
10. 

The base of the natural system of logarithms (also 
called Naperian or hyperboHc logarithms) is 6 = 
2.7182818284 .... 

A logarithm may be transformed from any given 
base to any other desired base by the relation: 

logaA^ 

logaft 

To transform a logarithm from base 10 to base e, 



4 ENGINEERING MATHEMATICS 

multiply by 2.302585 . . . (where 2.302585 . . . is the 
logarithm of 10 to the base e): 

logea = 2.302585 logio^ 

To transform a logarithm from base e to base lo, 

divide by 2.302585: 

logioa = ^ 3Q2535 ^^S^^ = 0.434294 logefl 

Special forms : 

log 1 =0 (to any base) 

loga a = I log^e = 1 

log = —QO log 00 = 00 

Cubic and Higher Degree Equations 

The approximate values of the real roots of an alge- 
braic equation containing only one variable may be 
found graphically. 

For instance, let it be required to solve the equation 
0(? + Ax — B = 0, This may be written sls o(? = 
— Ax+ Bj or as two simultaneous equations y = o(? 
and y = —Ax + B, The graph of each of these 
equations being plotted, the abscissas of their points 
of intersection give the real roots of the cubic. The 
curve y = o(^ should be plotted on cross-section paper 
by the aid of a table of cubes. The curve y = —Ax + 
B is the equation of a straight line, and is therefore 
determined by plotting two points. 

Algebraic equations of any degree may be solved by 
Newton's method of approximation; see page 39. 

Transcendental Equations 

The graphic method given under Cubic and Higher 
Degree Equations is also applicable to many trans- 



ALGEBRA S 

cendental equations. Thus, the equation Ax — sin :^ = 
may be solved by plotting the two simultaneous 
equations y = Ax and y = sin x. The curve y = sinx 
is readily plotted with the aid of a table of sines, while 
the other curve y = Ax is a straight Une passing 
through the origin. 

Infinite Series 

An infinite series is one containing an unlimited 
number of terms. Such a series is convergent if the 
sum of its terms is a finite quantity. It is divergent 
when the sum of its terms does not approach a finite 
limit. 

Comparison Test. A series is converging if each 
term in it is equal to or less than the corresponding 
term of a known converging series. 

Converging series for comparison: 

a-\- ar -\- ar^ -{• ar^ + • • • -\-ar^''^-\- • • • [^<1^ 
■+1 + 1+...+ • • 



1-2 ' 2-3 ' 3.4 ' ' w(«+ 1) ' 

l + ^ + f,+ •••+;^+ ••• [/»!] 

A series is diverging if each term in it is equal to or 
greater than the corresponding term of a known di- 
verging series. 

Diverging series for comparison: 
a+ ar + ar^ + ar^+ • • • + ar"""^ -|- . . . [r = 1] 
1+1+1 + 1 + 1+ . . . 

Ill 1 

l + - + i + i+ •••+-+... 



6 ENGINEERING MATHEMATICS 

Ratio Test. If, as the number of terms approaches 
infinity as its Umit, the ratio of the (n + l)th term to 
the nth term approaches some finite limit (a), the series 
is convergent if (a) is less than 1, divergent if (a) is 
greater than 1, and indeterminate by this method it 
(a) = 1. 

Oscillating Series. A series whose terms are alter- 
nately positive and negative is convergent if each term 
is numerically less than the preceding term. 

Standard Series 

/y^ /yO /y«4 

e^ = l + . + ^, + | + |;+... 

/yi /yO /yft 

e-=l-^ + 2j-3j + jj 

, /y^ /%/yO /yi 



1+1+1+1+ 
= 2.7182818 . . . 



_ limit / , ly 



/yi /yiO /y«4 

log(l + x)=x-| + J-|-+ . . . [lSiC>-l] 

/y»« /yO /yi. 

log(l-x) = -x-|-|-J- ... [1>«S-1] 

/yO /y^ /yl /yiV) 



ALGEBRA 



X^ . 0(^ 



X' 



T 



cosx - 1 2!~'~4! 6! "*" 8! 



11% 






' + ^^ A. 



2835 



cot a; = - 

X 



X 

3 



45 



2x' 



X' 



945 4725 






TT 



COS ^ X = - — sin~^ X 



2 



tan*"^ x = x — — + -F- 

sinhx = X + jj + || + yj' + ^ + 
COSh^=l+2;+4;+^+gj + 



X' 



5x^"^ 



X' 



• • [x2<7r2] 
. fl>x>-l] 



[1>X>-1] 



1 =J 



Complex Imaginary Quantities 

The imaginary unit = V- 

In representing complex Y 
imaginary quantities, it is 
usual to represent real 
quantities in the direction 
of the horizontal or X- 
axis, and imaginaries in 
the direction of the verti- 
cal or F-axis. Multipli- 




8 ENGINEERING MATHEMATICS 

cation by the imaginary unit, j, revolves a quantity 
through 90 degrees, in counter-clockwise direction. 

A complex number is the sum of a real and an 
imaginary, thus: 

A =a+jb ^a+V^b 
is a complex number. 

A complex number may be written in any of the 
following identical forms: 
^ = a+jb = r (cos0 + y sind) = re^ [6 in radians] 
a = r cos 6, 
b = r sin 6. 



in which 



The magnitude of the complex number, a + jb, is 

r = Va" + b^ 

Addition and Subtraction of complex quantities: 
To add two complex quantities, combine the real 
parts, and then the imaginaries, thus: 

(a +jb) + {c +jd) = {a+c) +j {b + d) 

In the same way, to subtract two complex quantities: 
{a+jb) - {c +jd) = {a-c)+j{b- d) 

Multiplication of complex quantities: 

To find the product of two complex numbers, mul- 
tiply out as in ordinary algebra, remembering that 
p = — 1, thus: 

{a +jb) (c+jd) = {ac - bd) +j (ad + be) 

Division of complex quantities: 
To divide two complex quantities, rationalize the 
denominator as follows: 

a+jb _ a+jb ^ c — jd _ (ac + bd) + j {bt — ad) 
c + jd c + jd c — jd c^ + d^ 



ALGEBRA 9 



Exponential Transformations 



smax = 



cos ax = 



2j 

^j'ax I ^—jax 



(e is the base of the hyperbolic logarithms; j equals 

De Moivre's Theorem: 

(cos +y sin Oy = cos nd+jsinnd 

Permutations and Combinations 

The number of permutations of n different things 

taken r at a time is 

nl 



Pr =n{n- I) . . . {n-r+ I) = 



{n — r)\ 

For n different things taken all at a time, the number 
of permutations is 

Pn = n{n-l) . . . (2) (1) =nl 

The number of permutations of n things taken all at 
a time, ni being alike, n-z alike, ns alike, etc., is 

P= — ... 

fill fhl fisl 

The number of combinations of n things taken r at 
a time is 

_n{n— I) . . . (n — r + 1) __ n\ 
*" r\ r\ {n^ r)\ 

For n things taken 1, 2, 3, . . . /^ at a time, the total 
number of combinations is 

C = 2^ - 1 



lO 



ENGINEERING MATHEMATICS 



GEOMETRY 

Plane Figures 



Right Triangle 

h = Vc2 — a2 



area 



_ 1 



a& 



Any Triangle 



_ 1 



area = ^bh 
area = Vs{s 



a) {s - h) {s 



c) 




Parallelogram 

area = ah 



Trapezoid 

area = \h{a + h) 



Regixlar Polygon 

area = J ahn 

n = number of sides 









GEOMETRY 



II 



Circle 

circumference = 2 Trr 

= wd 

area = irr^ 

(P 
= -4 




Sector of Circle 



arc = I = irr 



180° 



area = 4 r/ = wr^ 



360° 




Segment of Circle 

chord = c = 2 V 2 ^r - h^ 
area = | r/ — | c (r — /f) 




Parabola 

length of arc = 3-7 [v c (1 +c) + 

2.0326 logio (V^+ VI+"c)] 
in which 




Ellipse 

circumference = 



64-3 



Tr(a + b) 



fb-_aY 

[b-^aj 



64 



-K^:y 



(close approximation) 
area = irab 




12 



ENGINEERING MATHEMATICS 



SoUds 

Right Prism 
lateral surface = perimeter of base X h 
volume = area of base X h 



< 



> 



Pyramid 

lateral area = J perimeter of base X / 

h 
volume = area of base X -= 



Frustum of Pjn'amid 

lateral surface = \l(JP -\- p) 
P = perimeter of lower base 
p — perimeter of upper base 

volume = lh[A + a-{- V Aa] 
A = area of lower base 
^ a = area of upper base 



Right Circular Cylinder 
lateral surface = 2 irrh 

r = radius of base 
volume = irr^ 



Right Circular Cone 

lateral surface = irrl 

r = radius of base 






PLANE TRIGONOMETRY 



13 



Frustum of Right Circular Cone 
lateral surface = tt/ (i? + r) 
R = radius of lower base 
r = radius of upper base 
volume = iirh [R^ + Rr + r^] 



Sphere 



surface = ^tirr^ 
volume = ^irr^ 




Segment of Sphere 

volume of segment 



L 



to 



/ 



IP^X 



PLANE TRIGONOMETRY 

Right Triangle 



sin^ = 
tan^ = 



cos^ = 



cot ^ = - 
a 



sec ^ =7 




sin A = cos 



cosec^ = 




a 



cos 






14 



ENGINEERING MATHEMATICS 



tan A =- cot I - — ^ 



cot^ = tan (^-a)^- ^^^(|+ a 
A = secf^- ^j= - sec\^+ Aj 



sec A = cosec 



cosec 



sin {^ A) = — sin A 
tan(— A) = — taxiA 
sec (— A) = sec A 



cos (— A) = cos A 
cot {— A) = — cot A 
cosec (— A) = — cosec i4 







NUMERICAL VALUES 




Angle . . 


0° 


30° 


45° 


60° 


90° 


sin 





1 

2 


V2 
2 


Vs 

2 


1 


cos 


1 


V3 
2 


V2 
2 


1 

2 





tan 





V3 
3 


1 


Vs 


00 


cot 


00 


Vs 


1 


Vs 
3 






Trigonometric Formulae 



tanx = 



sec:i: = 



tan jc = 



sin:;i[; 
cos:r 

1 
cosx 

1 

cot:x: 



cotx = 



cosx 
sinx 

1 



cosec X = 



cotx = 



smo; 
1 

tan jc 



PLANE TRIGONOMETRY 1 5 

sin^ X + cos^ X = I 

sec^ X = 1 + tan^ x 

cosec^ X = 1 + cot^ X 

sin {x + y) = sin x cos y + cos x sin 3; 

cos (x + y) = cos :^ cos y — sin x sin y 

, , V tan X + tan y 

tan (x + y) = z 7 r-^ 

1 — tan X tan y 

^ , , X cot X cot 3; — 1 

cot (^ + }') = — I — r^S: — 
cot X + cot 3^ 

sin (x — 3^) = sin X cos y — cos x sin y 

cos (x — 3^) = cos X cos 3; + sin X sin y 

tan X — tan 3; 



tan (x — y) = 
cot (x — y) = 



1 + tan X tan y 
cot X cot 3;+ 1 



cot y — cot X 
sin 2 X = 2 sin X cos x 
cos 2 X = cos^ X ~ sin^ x 
2 tanx 



tan2x = 
cot 2 X = 



1 — tan^ X 
cot^ X — 1 



2 cotx 
sinf 



.-si'- 



1 . / - ~ cos X 



+ cosx 
cos 



tan|x = 



2 

1 — cos X 



sinx 

sin X + sin y = 2 sin i^ (x + 3;) cos | (x — y) 
sin X — sin y = 2 cos | (x + y) sin J (x — y) 
cos X + cos y = 2 cos I (x + y) cos hi^-^ y) 
cos X — cos y = — 2 sin I (x + y) sin | (x — y) 



i6 



ENGINEERING MATHEMATICS 



a 



Solution of Any Plane Triangle 

b c 



sin A sin B sin C 

a^ ^b^^ c^ — 2bc cos A 
a-b ^ tan ^ {A - B) 
a + b~ tan^iA + B) 

a sin C 



tSLYiA == 




b — a cos C 



SPHERICAL TRIGONOMETRY 
Right Spherical Triangles 

cos c = cosa cos b 
sin a = sin c sin A 
sin b = sin c sin 5 
cos A = cos a sin 5 
cos B = cos i sin ^ 
cos A = tan b cot c 
cos B = tan a cot c 
sin & = tan a cot A 
sin a = tan b cot B 
cos (; = cot A cot 5 

Oblique Spherical Triangles 




i 



sma 



sin 6 



smc 



sin ^ sin 5 sin C 
cos a = cos&cosc + sin6sin(;cos^ 
cos ^ = sin5sinCcosa— cos5co^C 
cot a sin 6 = cot-4 sin C+cos C cos 6 ^ 
s = ^{a + b + c) 




SPHERICAL TRIGONOMETRY 17 

sin (i) = J sin(5-^,)sin(.- g 
\2 / V sin b sm c 

M \ 4 /sin ^ sin (5 — a) 

cos h^ = V ^^T^ 

\2/ V sin 6 sine 

tan (-] = 4 AiJ^ (^ - b) sin (5 - c) 
\ 2 / V sin 5 sin (^ — a) 

. /a\ . / cos S cos (5 — ^) 

^^" (2 j = V - 



sin ^ sin C 



it)-^' 



^ _ ^ , cos (5 — B) cos (5 — C) 



sin ^ sin C 



©=v/ 



, - . . , COS 5 COS (S — A) 

tan ' » — * ' 



cos (5-^) cos (5- C) 



^ 1 , - . sin n^ - ^) ^ 1 
tan^(a-&)=^.^,^^_^^^tan|c 

1 / . 7 \ cos h (A — B) . 
tan^(a+5)== ^^^,^^^^) tan^. 

tan H^ - ^) = 'l^'I^^'lS c<^t I C 
sm I (a + 6) 

tani (A + B)= ^^4fe|! cotK 
cos I (a + 6) 

1 sinH^ + B) tSLiij (a - b) 

tan ^ c = ; — J— r-j ^T 

sm^ (A — B) 

Application of Spherical Trigonometry to 
Navigation 

To find the shortest distance between two points on 
the earth's surface and the bearing of each from the 
other, the latitude and longitude of each being given. 
(From W. A. Granville's '^ Plane and Spherical Trigo- 
nometry.") 



l3 ENGINEERING MATHEMATICS 

(i) Subtract the latitude of each place algebraically 
from 90°, taking North latitudes as positive and South 
latitudes as negative. The results will be the two sides 
of a spherical triangle. 

(2) Find the difference of longitude of the two 
places by subtracting the lesser longitude from the 
greater if both are East or both are West; but adding 
the two if one is East and the other West. This gives 
the included angle of the triangle. If the difference of 
longitude found is greater than 180°, then subtract it 
from 360° and use the remainder as the included angle. 

(3) Solving the triangle by the formulae for tan | 
{A — B)j tan ^ {A + B), and tan | c, the third side 
gives the shortest distance between the two points in 
degrees of arc, and the angles give the bearings. The 
number of minutes in the arc will be the distance 
between the places in nautical miles. 

Illustration. Find the shortest distance along the 
earth's surface between Boston (latitude 42°21'N., 



Nojth Pole 




South Pole 



longitude 71° 4' W.) and Capetown (latitude 33^56' 
S., longitude 18° 26' E.) and the bearing of each city 
from the other. 

(1) a = 90°- 42° 21' = 47° 39' 

J = 90° - (-33° 56') = 123° 56' 



PLANE ANALYTIC GEOMETRY 



19 



(2) C = 71° 4' + 18° 26' = 89° 30' = difference in 

longitude. 

(3) Solving the triangle as explained above, we get 

c = 68° 14' = 68.23° = 4094 nautical miles. 

A = 52° 43' = bearing of Boston from Cape- 
town. 

B = 116° 43' = bearing of Capetown from 
Boston. 



PLANE ANALYTIC GEOMETRY 
The Straight Line 



I. The slope equation: 

y = MX + b 
m = slope = tan^ 
b = intercept on F-axis 



n. The intercept equa- 
tion: 






where a and b are the inter- 
cepts on the X and F-axes. 




III. Line through the 
points (x',/) and {x'\y"): 
y — y' X — x' 



y" - y' 



X 



X 



(x',y') 



(x".y") 



•X 



20 



ENGINEERING MATHEMATICS 

Y 



IV. Line through the 
point {%', y), with slope m: 

y ^ y =^ m{x — x') 




V. Distance from the point {x',y') to the line 
Ax + By + C = Q: 

Ax' + 5/ + C 



d = 



± V^2 4. ^2 



VI. Distance between the points {x', y') and 
rf = V(x' - x'y +{y' - y"y 



Transformation from Rectangular to Polar 
Coordinates 

X = r cosd 
y = rsind 

r = radius vector = Vx^ + y^ 



= polar angle = tan~- 
r Vx^ + / 



X 




PLANE ANALYTIC GEOMETRY 



21 



The Circle 



I. Circle of radius r with cen- 
ter at origin: 

^ + y^ = y^ 




II. Circle of radius r 
with its center at the 
point (a, b) : 




*— a 



III. Tangent at the point (a, 6) of the circle o^ + 

y2 _ ^2 jg 

ax-\-hy ^ r^ 

IV. Slope equation of the tangent to the circle 

iX^ -f y2 _ ^2 Jg 

y = mx zh r v w^ + 1 



V. Polar equation of 

circle of radius a passing 
through the origin, and 
having its center on the 
X-axis: 

r = 2 a cos 




22 



ENGINEERING MATHEMATICS 




VI. Polar equation of 

circle of radius a passing 
through the origin, and 
having its center on the 
F-axis: 

r = 2 a sin 



Parabola 

Definition. The parabola is the curve generated by 
a point moving so as to remain always equidistant from 
a given fixed point and a given fixed line. 

The fixed point is called the focus ; the fixed line is 
called the directrix. 

I. Parabola with its v 
axis along the X-axis and 
vertex at origin: 

y^ = 4:ax 

where a is the distance 
from the origin to the 
focus. 

II. Parabola having its 
axis along the F-axis and 
vertex at origin: 

x^ = 4:ay 

where a is the distance 
from the origin to the 
focus. ~ 



III. General equation of a parabola with axis 
parallel to the Z-axis: 

X = ay^+ by + c 



PLANE ANALYTIC GEOMETRY 
the vertex is at the point 



23 



/ P-4ac _ _6_\ 
V 4a ' 2 a) 



IV. General equation of a parabola with axis par- 
allel to the F-axis: 

y = ax^+ bx + c 

the vertex is at the point 

/ _&_ b^-4:ac \ 
[ 2a' 4a / 

V. Slope equation of the tangent to the parabola 
y^ = 4 ax is 

y = mx H — 

VI. Slope equation of the tangent to the parabola 
^2 = 4 ay is 

y = mx — arr? 

Ellipse 

Definition. The ellipse is the curve generated by a 
point moving so that the sum of its distances from two 
fixed points is always constant. The fixed points are 
called the foci. 

I. Equation of el- 
lipse with center at 
origin: 

i ^^ _i_ 3^^ _ 1 

j where a and h are one- 
half the major and 
minor axes. 




24 



ENGINEERING MATHEMATICS 



II. Slope equation of the tangent to the ellipse 



-^ + 7? = 1 IS 



a^ 



62 



Hyperbola 

Definition. The hyperbola is the curve generated 
by a point moving so that the difference of its distances 
from two fixed points is always constant. 

I. Equation of hyperbola with center at origin: 




n. Equation of conjugate hyperbola: 



x^ y^ 



a^ ¥ 



= -1 




><^ 



PLANE ANALYTIC GEOMETRY 25 

III. Equations of asymptotes of the hyperbola 



-:; — 7^ = 1 are 



b 

y = -X 
a 



b 

— -X 

a 



IV. Slope equation of the tangent to the hyperbola 

x^ f . 

"2 -^ = 1 IS 

y = mx dz ^c^yr^ — b^ 



V. Slope equation of the tangent to the conjugate 
hyperbola ^ - ^ = - 1 is 



y = mx zb V&2 — a%2 

Cycloid 

Definition. The cycloid is the curve generated by 
a point on the circumference of a circle as the circle 
rolls along a straight line. 




a: = a (0 ~ sin 0) 
y = a (1 — COS0) 



ri^-V2 



X = a vers~" ' — v z ay 
a 



where a is the radius of the rolling circle. 



26 



ENGINEERING MATHEMATICS 



Epicycloid 
Definition. The epicycloid is the curve generated 
by a fixed point on the circumference of a circle which 
rolls externally on the circumference of a fixed circle. 




X =^ {a+ b) COS0 — b cos ( — 7 — dj 
y = {a+ b) smd - bsm(^^ e) 

where a is the radius of the fixed circle, and b the radius 
of the rolling circle. 

Hypocycloid 
Definition. The hypocycloid is the curve generated 
by a point on a circle which rolls internally along the 
circumference of a fixed circle. 




X = (a— b) cosd + b cos f — t~^) 



SOLID ANALYTIC GEOMETRY 



27 



y = (a — b) sind — b sin ( — ; — 6 



(^'') 



where a is the radius of the fixed circle and b the radius 
of the rolling circle. 

The Catenary 

The catenary is the curve which a heavy cord or 
perfectly flexible chain of uniform density forms, due 




to its own weight, when freely suspended between two. 
points. 



a 



y = 2 v^'' + ^ 



) = a cosh 



SOLID ANALYTIC GEOMETRY 

The direction cosines of a line in space passing 
through the origin are the cosines of the angles which 
the line makes with the rectangular coordinate axes. 
The direction cosines of any line in space are the 
direction cosines of a line parallel to it and passing 
through the origin. 



28 



ENGINEERING MATHEMATICS 



I. Distance from the point {x, y, z) to the origin; 

r = Vx^ + / + ^2 
z 




II. The direction cosines of the line from the point 
(x, y^z) to the origin are: 



(x,y,z) 




cos a = - = 



i)C 



r Vx2 + / + 



>^^-i-s^ 



COSjS 



_3^_ 



r Vx2 + / + z2 



cos 7 = - = 



r V:r2 +3,2+22 

III. The sum of the squares of the direction 
cosines of a line is equal to 1, 

cos2 a + cos2 /3 + cos2 7=1 



SOLID ANALYTIC GEOMETRY 



29 



IV. Distance between the points {x, y, z) and 

d = V{x- x'Y +{y- y'Y +{z- z'f 

V. Direction cosines of a line joining the points 

(x, y, z) and {x\ y\ z')\ 



X — X 



cos a = 



COSjS = 



cos 7 = 



«/v »a/ 



d V(x - xy +{y- yy +(z- zy 

y-y _ y-y 

d V(x - x'f +{y- yy + (2 - zy 

Z— Z^ 4^ Z— Z^ 



d 



V{x - xy +{y- yy +(z- zy 

VI. The angle between two Unas in terms of their 
direction cosines: 

cos ^ = cos a cos a + cos ^ cos jS' + cos 7 cos 7' 



Vll. Intercept equa- 
tion of a plane: 

a b c 




where a, &, and c are the 
intercepts of the plane 
on the Z, Y , smd Z 
axes. 



VIII. General equation of a plane : 

Ax + By + Cz+D = 

IX. Distance from the point (x\ y, z') to the plane 
Ax-\-By^Cz^B = ^\ 

Ax' + By ^Cz' ^D 



d = 



± Va^ + 52 + C2 



30 ENGINEERING MATHEMATICS 

X. Straight line through the two points {x'\ y'^ s'O 
and {x\ y\ z') : 



X — 


x' 


= 


y. 

y" 


-y 

-y' 


= 


z — 


z' 


x" - 


■x' 


z"- 


-z' 



XI. Straight line through the point {%', y', z')» and 
making the angles a, jS, and 7 with the coordinate axes: 

X— x' _y — y' _z— z^ 
cos a cos /5 cos 7 

XII. General equation of a straight line is given by 
the equations of two intersecting planes: 

A'x+ B'y+ Cz+ D' = 
A''x + B"y + C'z + D" = Q 

CALCULUS 
Application of Differential Calculus 

The following list includes some of the principal 
formulae necessary for the solution of geometrical and 
physical problems, relating to any curve y = f (x). 

Rectangular Coordinates : 

dv 
Slope of the tangent at the point {x, y) = -^ 

dx 
Slope of the normal = — j- 

Equation of the tangent at the point {xo, y©), Xo and 
yo being the coordinates of the given point, is 

dyo r V 

Equation of the normal at (xo, y^ is 



CALCULUS 31 

dx 
The intercept of the tangent on the Z-axis hk — y — 

The intercept of the tangent on the F-axis \sy — x-~ 

ax 

The intercept of the normal on the Z-axis \^x-\- y-^ 

dx 
The intercept of the normal on the F-axis is 3; + ^ t- 

Length of the tangent from its point of contact with 
the curve to the Z-axis is 



M^ 



Length of the tangent from its point of contact with 
the curve to the F-axis is 



V'+(D* 



Length of the normal from its point of contact with 
the curve to the X-axis is 



\/'+(i)' 



Length of the normal from its point of contact with 
the curve to the F-axis is 



V'+(|J 



dx 
Length of the sub tangent = y-j- 

dy 
Length of the subnormal = J-r- 

Differential length of the arc = ds = V(dxy + (dy)^ 



32 ENGINEERING MATHEMATICS 



Radius of curvature = 



h( 



dySff 
dx) \ 



d^ 



Curvature is the reciprocal of radius of curvature. 
Length of the perpendicular from the origin on the 
tangent (to the curve) is 

dy 



Polar Coordinates : 

tan \l^ = r -r-, where ^ is the angle between the radius 

vector and that part of the tangent to the curve at 
{r, 6) drawn back toward the initial line. 

Length of polar sub tangent = ^^ ir 

dr 
Length of polar subnormal = -jz 

uu 

Differential length of arc = ds = V(dry + r^ {dSf 



Length of the perpendicular from the pole on the 
tangent = p = r^-ri also, 

1 = 1 1 1/^V 



A 



"^ 



CALCULUS 33 

Formulae of Differential Calculus 

d iau) = adu 
d{u + v) = du-{- dv 
d (uv) = vdu + udv- 
J /^\ _vdu — udv 

d (x"^) = nx"^'^ dx 

d (x^) = yx^-^ dx + x^ \oge x dy 

d (e^) = e^dx 

d {a"") = a^ loge a du 

d (loge x) = -dx 

X 

d (sin x) = cos x dx 
d (cos x) = — sin x dx 
d (tan x) = sec^ x dx 
d (cot x) = — cosec^ x dx 
d (sec x) = sec x tan x dx 
d (cosec x) = — cosec x cot x dx 
dx 



d (sin~^ x) 



d (cos""^ x) == — 

d (tan~^ x) = 
d (cot~^ x) = 
d{sec~^x) = 



Vl-x^ 
dx 



Vl-x' 
dx 



l + x^ 
dx 



1 + x^ 
dx 



x \^x^ — 1 
dx 



d (cosec"^ x) = — / . 

^ Vr2 _ 1 



34 ENGINEERING MATHEMATICS 



Maxima and Minima 

The maximum or minimum values of a given func- 
tion y = / (x) are obtained as follows: 

(1) Find the first derivative -j- and equate it to zero. 

(2) Solve the resulting equation for values of x. 

(3) In order to determine whether these values of 
X make y maximum or minimum, obtain the second 

derivative -7^ of the given function. 

d^y 

(4) Substitute separately in the expression for ~~^ 

each of the values of x found above. Values of x that 

d^y , , 
make -7^ positive correspond to minimum values of 

d^y 
the function, and values of x that make -7^2 negative 

correspond to maximum values of the function. 

(5) Substituting these values of x in the given 
function y = f {x)y we obtain the maximum or minimum 
values of y. 

Illustrative Example. Find the values of x which 
will make the function y = 6x-\-3 x^— 4:X?sl maxi- 
mum or a minimum, and find the corresponding values 
of the function y. 

(1) The first derivative of 3/ is 

^ = 6+6x-12x2 

(2) The values of x which make y maximum or 

dy 
minimum will make ~f- = Oj therefore 

6+ 6x— 12x^ = 0, or x^ — |x = | 



CALCULUS 35 

solving, x = Jzbf = +l or — | 

Hence, the maximum or minimum values of y must 

occur when x = 1 or — |. 

(3) To determine whether these values are maxima 
or minima, we obtain the second derivative of y\ thus: 

(4) When x = 1, -j^ = — 18, which corresponds to 
a maximum value of y. 

When X = — i, -7^ = +18, which corresponds to a 

minimum value of y, 

(5) Substituting these values of x in the given func- 
tion, we have 

when x = l, ;y = 6+3 — 4 = 5, a maximum 
when X = — |, y = — 3 + f + | = — |, a minimum 



Taylor's and Maclaurin's Series 
Taylor's Series: 

where / denotes the function, /' the first derivative, f 
the second derivative, etc. 
Maclaurin's Series: 

/(x) =/(o)4-/(o)f +r (o)|J+r(o)^+ . . . 

where / (0) denotes the value of the function when is 
substituted for x, j' (0) the value of the first derivative 
when is substituted for x, etc. 



36 ENGINEERING MATHEMATICS 

APPLICATION OF INTEGRAL CALCULUS 
Lengths of Curves 
Rectangular Coordinates : 



y 1 +[-f-] dx 



From the equation of the given curve, find y in terniL 

of x: then differentiate in order to obtain -r-, and 

ax- 
substitute its value in the formula. The lower limit 
a is the initial value of x, and the upper limit b the final 
value of X. 

Or, similarly, by solving for x in terms of y, and 

dx 
obtaining -^, the length of the curve is given by the 

formula 



'J:mw^' 



where c and d are the initial and final values of y. 
Polar Coordinates : 



length of curve = 5=1 V/l + r^ ( — j dr 

where a and b are the limiting values of r. 
Or, 



length of curve = 5=1 V/ r^ ~^ \^) ^ 
where 6^ and 6^^ are the limiting values of 6. 



CALCULUS 37 

Plane Areas 
Rectangular Coordinates : 

The area included between a curve, the Z-axis, and 



the vertical Hnes x = a and x = b is 

= A = I ydx 

tJ a 



area 



The value of y in terms of x is found from the given 
equation and substituted in the formula. The initial 
value of X is a, and the final value b. 

Similarly, the area included between a curve, the F- 
axis, and the horizontal lines y = c and 3/ = J is 



area 



= A = j xdy 



where c and d are the limits of y. 

Polar Coordinates : 

The area included between a given curve and two 

given radii is 

re" 
area = A = ^ I r^ dd 

where B" and B^ are the limiting values of B. 

Areas of Surfaces of Revolution 

For revolution about the X-axis, 



area 



A.2.£y^l + {fjd, 



where the value of (-1^) is found from the given equa- 
tion. The initial value of x is a, and the final value b. 



38 ENGINEERING MATHEMATICS 

For revolution about the F-axis, 

area = A = lir \ ^\ ^-\-\-r\ ^V 
where c and d are the limiting values of y. 

Volumes of Solids of Revolution 
Rectangular Coordinates : 

volume = Fx = TT I y^ dx 

is the formula for the volume generated by revolving 
the given curve about the X-axis. The limiting values 
of X are a and b. 

Similarly, the volume generated by revolving the 
plane figure about the F-axis equals 



Vy = T I x^ dy 



where c and d are the initial and final values of y. 
Polar Coordinates : ' 

When the plane figure is revolved about the Z-axis, 

the volume generated is 

Va: = 2T C ir'^sinddddr 

For revolution about the F-axis, the volume generated 
is 

r^ cos 6 do dr 



• '\ff' 



INDETERMINATE FORMS 

f (x) 
If the fraction v. \ \ gives rise to the indeterminate 

form TT or — , when x approaches a as a limit, the in- 



CALCULUS 39 

determinate form may be replaced by a new fraction, 

f (x) 

•L/ / X , the numerator of which is equal to the deriva- 

F {x) 

tive of the given numerator, and the new denominator 

is equal to the derivative of the given denominator. 

The value of this new fraction, as x approaches a, is the 

limiting value of the given fraction. If this again 

becomes indeterminate, it may be necessary to repeat 

the process several times. 

Example. Find the limiting value, when x = l, ot 

the fraction 

x^ + x-2 

x^- I 

f(x) x^ + x-2 ^ 

t^T^ = ^ z — = ^, when x = 1 

F {x) x^ — 1 

fix) 2x+l 3 , ^ 

F {x) 2x 2 

Hence, the required limiting value is f . 



SOLUTION OF EQUATIONS 

Algebraic equations may be solved by Newton's 
method of approximation. Thus, let it be required to 
solve an equation of the form Ax^ + Bx^ + Cx = D. 
Find, by trial, a number, r, nearly equal to the root 
sought, and letr -\- h denote the exact value of the root, 
where his sl small quantity the value of which must be 
determined. Substituting r + ^ for x in the given 
equation and neglecting all powers of /j higher than the 
first, we have, approximately, 

J Ar^ + Br^ + Cr - D 
-3Ar^-2Br-C 

It will be observed that the numerator of the above 



40 ENCxINEERING MATHEMATICS 

fraction is the first member of the given equation after 
D has been transposed and x changed to r, and the 
denominator is the first derivative of the numerator 
with its sign reversed. The correction h added, with 
its proper sign, to the assumed root r, gives a closer 
approximation to the value of x. Repeat the opera- 
tion with the corrected value of r, and a second correc- 
tion will be obtained which will give a nearer value 
of the root; two corrections generally give sufficient 
accuracy. 
Illustration. Find a root of the equation 

The value of /? is 

r^ -f- 2 /^ + 3 r - 50 



h = 



-3r^- 4:r 



By trial, we find that x is nearly equal to 3. On sub- 
stituting 3 for r, we have 

2 
/? = — — = — 0.1, approximately 

Hence, x = 2.9, nearly. If we substitute this new 
value of r, the new value of h equals +0.00228. 
Hence x = 2.90228. If we repeat the operation with 
this last value of r, the value of h is then found to be 
+0.0000034. Hence x = 2.9022834. 

CURVE TRACING 

The usual method of tracing curves consists in 
assigning a series of different values to one of the 
variables, and calculating the corresponding series of 
values of the other, thus determining a definite number 
of points on the curve. By drawing a curve through 



CALCULUS 41 

these points, we obtain a graphical representation of 
the given equation. 

The general form and peculiarities of the curve 
can be easily determined and sketched by the following 
steps: 

(1) If possible, solve the equation of the given curve 
for one of its variables, y for example. If the equation 
then contains only even powers of x, it is symmetrical 
with the F-axis. 

Or if, when solved for x, it contains only even powers 
of y, it is symmetrical with the X-axis. 

(2) Find the points in which the curve cuts the axes 
by solving the equation of the given curve in turn with 
the equations x = and y = 0. 

(3) Find the values of x^ if any, which make y 
infinite; similarly, test for infinite values of :^. 

(4) Find the value of the first derivative ~-\ and 

ax 

thence deduce the maximum and minimum points of 

the curve. 

In tracing polar curves, write the equation, if 

possible, in the form r = f {d)\ and give d such values 

as make r easily found, as for example, 0, i tt, tt, | tt, etc. 

dr 
Putting 1^ = 0, we find the values of Q for which r is 

a maximum or minimimi. 



METHODS OF INTEGRATION 

(By parts, substitution, etc.) 

When the numerator of a fraction contains a variable 
to an equal or a higher power than the denominator, 
the fraction must be reduced to a mixed quantity (by 



42 - ENGINEERING MATHEMATICS 

actually dividing the denominator into the numerator) 
before it can be integrated. 

If an expression cannot be integrated by the formulae 
given in the table of integrals, one of the following 
methods may be used to obtain a solution. 

Partial Fractions 

A fraction may be resolved into partial fractions, 
which can be integrated separately. 
Example. To integrate 

dx 

{x + a) (x + b) 

Let 

1 ^ A B 

{x + a) ix +b)~ {x+a)'^ {x+b) 

where we must determine A and B. 

Clearing of fractions, 

\ = A {x+ b) + B {x+ a) =^ {A + B) x^- {bA + aB) 
The coefficients of like powers of x on both sides of the 
equation are equal; therefore, 

A + B = 
bA + aB ^\ 

whence A = f and B = 7 

— a a — 

and 



J (x+a){x+b)'^''~J (x+a)'^^+J (x+ft)"^^ 

These forms are now integrable by the table of integrals, 
the result being 

/I 1 1 
/ , w , ,sd x = 7 log(x+a)H Aog(x+b)+C 
{x+a){x+b) b—a a — b °^ 

where C is the constant of integration. 



CALCULUS 43 



Integration by Parts 

To integrate by parts, apply the formula 



I udv = uv — j vdu 



The method of integration by parts is most effective 
in dealing with the integration of products, involving 
logarithms, and trigonometric and inverse circular 
functions. 

Generally, the most complicated quantity which can 
be integrated directly by one of the fundamental for- 
mulae (see Table of Integrals, page 46) is equated, 
with the differential, to dv, and the remaining part is 
equated to u. 

Example. To find 

X log (x) dx 



I' 

Let u = log X and dv — xdx 

dx i oc^ 

then du ^ — v = I xdx = -^ 

X . J 2 

Substituting in the formula 

\ udv = uv— \ vdu 

we have 

/-v" ioc^ dx 
X log (x) dx = \og{x) . 2" "" J 2" ^ 

= |-log(x)-|+C 

Integration by Substitution 

I. Differentials containing fractional powers of x 
may be integrated by the substitution 

X = 2" 



44 ENGINEERING MATHEMATICS 

where n is the least common denominator of the frac- 
tional exponents of x. 

II. Expressions involving only fractional powers of 
{a + bx) may be rationalized by the substitution 

{a + bx) = z^ 

where n is the least common denominator of the frac- 
tional exponents of (a -\- bx). 

III. To integrate expressions containing 

V :x:^ + ax + &, 
use the substitution 

V x^ -\-ax-\'b = z — x 

IV. Expressions containing V— ^^ + ax + 6 may 
be rationalized by the substitution 

v— x^+ ax+ b = {x — 6) z 

where (x — 6) is sl factor of (— x^ + ax + b). 

V. A differential containing sinx and cosx can be 
transformed by means of the substitution 

^ X 
tan - = s 

from which 

2 z ^ 1-^ ^ 2dz 

sm X = T—. — ^ cos X = T— i — o dx = :i— ; — ^ 

1 + 2^ 1 + r 1 + 2^ 

VI. A very useful substitution is 

1 

X = - 

2 

VII. Differentials involving Va^ — x^ may be ra- 
tionalized by the substitution 

X = a sin ^ 

VIII. Differentials involving Va^ + x^ may be ra- 
tionalized by the substitution 

X = a tan d 



CALCULUS 45 

IX. Diflerentials involving Vx^ — c? may be ra- 
tionalized by the substitution 

X = a sec ^ 

Reduction Formulae 

The purpose of the following reduction formulae is to 
simplify an integral of the form 



/ 



/ 






x"^ (a + bx^ydx - , ^ , , , X ,, 

{np -\- m+ l)b 

(m — n-\- V) a C , , , . , 

{np + w + 1) 6 J 

This formula enables us to lower the exponent of x 
by n, without affecting the exponent of (a + bx^). 
Method fails when {np + w + 1) = 0. 

II. fx-^ {a + bx-y dx = ^7""' ^^ + ^"^^^"^ 
J {np + m+ I) 

+ -' . . ,x / x'^{a+ bx'^y-^ dx 

{np -\- m+ i) J 

By this formula, the exponent of {a + bx^) is lowered 
by 1, without affecting the exponent of x. 
Method fails when {np + w + 1) = 0. 

III. / x^ (a + bx'^y dx = r — hrv^ — 

J {m+ l)a 

{m+ 1) a J 

By this formula, the exponent of x is increased by n^ 
without affecting the exponent of {a + bx""). 
Method fails when m = —1. 



46 ENGINEERING MATHEMATICS 

IV. A;- (a + bx-y dx = _ ^"^'(^+M^+' 
J n{p+ l)a 

, (np + n+ m+ 1) r , . , ^ ^. ^ 

n{p+ l)a J 

This formula enables us to increase the exponent of 
(a + bx"") by 1, without affecting the exponent of x. 
Method fails when ^ = — 1. 

TABLE OF INTEGRALS 
Fundamental Forms 






x^dx = 



jL»n+l 



n+l 



'dx , 
— ^logx 



e^ dx = (? 

a^ dx 



a" 



\ogea 
dx 



l + x'' 
dx 



= tan~^ X 



= sin~^ X 



Vl- x^ 

dr. 

sec~^ X 



X Vx^ — 1 

smxdx = — cosx 






cos xdx = sin x 

tan xdx = log (sec x) 

cot xdx = log (sin x) 



TABLE OF INTEGRALS 47 

/ sec xdx = log tan (9 + 2;) 
/ cosec xdx = log f tan - J 
tan X sec xdx =^ sec x 



f 
f 
f 



cot :jc cosec X (/x = — cosec x 
sec^ xdx = tan x 
cosec^ xdx = — cot x 



Expressions involving {a + bx) : 
= 7 log {a + bx) 



{a + bx) b 

dx 1 



{a + bxY b{a-\- bx) 

= Tk[(^ + bx — a\og {a -\- bx)] 



f. 

J (a + bx) ~ W- 



dx 1^ a-\-bx 

log 



x{a-\- bx) a x 



dx 1 \ . a+bx 



x{a-\r- bx)^ a{a + bx) c? x 



dx \ , b , a-\-bx 

-— + 72 log 



x^ {a + bx) ax a^ X 

Expressions involving (a + bx^) or (a^ zb x^) 






rfx 1 ,x 

= -tan~^- 



a^ + ^^ d d 

dx 1 , a + ^ 

= :^log 



a^ — x^ 2 a a — X 



48 ENGINEERING MATHEMATICS 

r dx 1 , Va+xV-b ., ^f, ,^rt 

or / , , o = — / log —7^ 7= ifa>0, J<0 

J a + bx^ 2V-ab Va-xV^ 

/dx X 1 r dx 

(a + bx^^y ~ 2a(a + bx') '^iTaj a + bx^ 

J a+ bx^ 2b \ bj 

/dx _ 1 , x^ 

X {a-\- bx^) 2a a+ bx^ 

Expressions involving V a + bx: 
I V a + bxdx = Yl ^(^ + ^^y 

j x"va 

jxWa 



-\-bxdx = 



2{2a- 3bx)V{a+bxY 



15 b^ 



-\-bxdx = 



2 (8 a2 - 12 a&x+ 15 bH'') V(a + bxf 



105 63 

dx = 2 \/a -\-bx-\- a j — . 

^ J X \ a + bx 

/dx _ 2 V a + bx 
Va + bx b 

/' xdx 2 (2 a— bx) / — r^— 
/ = —TT2 -^cL + bx 

Va + bx 3 b^ 

r x^dx 2(Sa^-4abx+3b^x^) /—t-t 

1-7= = , g.3 ^-Va + bx 

J Va+bx 156^ 

/ dx _ J_ Wa +bx— Vg ] 
X Va+bx Va \Va + bx + Va\ 



or 



/dx 2 ^ .. a+bx 
— , = . — tan-i V 

xVa+bx v-a ^ ""^ 



TABLE OF INTEGRALS 49 

/ dx __ Va+bx _ Jb_ C dx 
x'^ V a -\- bx ^^ 2 a ,/ ^ V ^ _|_ i)x 

Expressions involving V a^ — x^ or V a^ + x^: 

I Va^ — x^ dx = -\x Va^ — x^+ a^ sin~^ - 

/dx , , X 

Va^ - x^ ^ 

r dx _ _ 1 [1 ^+Va[±^l 
J X Va2 zt ^2 ~ <z L ^ a; J 

/^^.. = V^^^- a log [^+^^] 



±x? 



fv{a^-x^f dx=^{5a''-2 x^) V^F^^^ + ^a* sin-i - 

t7 o o Q' 

J x^\/a^ — x^dx = — ^ V (a^ — x^Y 

d^ I / OCl 

+ -^ he Va^ — x^ + a^ sin~^ - 

/x dx X ^ /~o o I ^ • 1 *^ 

; = — 7^ V a^ — :r2 + — sin-i - 

V a^ - ^^2 2 2 a 

/^^ Va^ - x^ 



x^ Va^ — x^ a^ 



a'-x 

^ ^ V a^ — rx:^ . - at; 

dx = sui"^ - 

X'' X a 



/Va^ — X 
x^ 

Expressions involving V x^ + a^ or V x^ — a^: 

jVx^±a^dx = i [xVx2±a2±:a21og (x + Vx^ i a^)] 



50 ENGINEERING MATHEMATICS 

I , ^^ = log {x + Va;2±a2] 






dx 1 , a 



cos~^ 



Vx2 - a^ a X 



2 -1 ^ 



J X 

rV(x2 ± a2)3 </x = I (2 «2 _t 5 ^2) V^^^i^ 



+ ^log(A; + Vr»;2±o2) 

O 



Jx dzo; 



jxWx^zLa'dx = I (2 x2 ziz a^) V:^2 j_ ^2 



8 



*/ V :;c2 dz a^ ^ ^ 

/ dx _ Vx^ =b g^ 

-^ • = + I0g(x + Vx2=ba2) 

Expressions involving Vzbax^ -\- bx + c: 

f , ^^ = ~ log (2 ax+^^+ 2 Va Vax^+te+^j 

*^ Vax^+bx+c Va 



TABLE OF INTEGRALS 5 1 

/\/ ax^ -\rhx-\- cdx — Vax^ -\-hx-\- c 
4a 

W'-^ac r dx 

Sa J V'ax^ + bx+c 

/dx _ I , _^ / 2 ax — b \ 

V-ax^+bx+c ~ Va ^^^ Wb^ + 4 aJ 

/V—ax^ + bx+ cdx = V— ax^+ bx+ c 
4:a 

P+4:ac r 

"a J 

Formulae involving v 2 ax — x^\ 

's/lax — x^ dx ■= — - — V 2 ax— o(?- -\- 1^ sin~^ 

2 la 

// 3 (m ~\~ ax — 2 X / 
X V 2 ax— x^ dx = 7 V 2 ax — x? 



b^+4ac r dx 

8^ J V- ax^ + bx + c 



, a^ .X 

2 a 



dx ,x 



= vers~^ 



JVlax-x^ '^^^ ^ 

/ . — = — V2 ax — x^ + a vers"^ - 
V 2 ax - x2 ^ 

/rfx V 2 ax — x^ 



X ■\/2 ax - x2 ax 



' dx = V 2 ax — x^ + a vers~^ - 

X <z 

/Jx _ X — a 

V(2 ax - x2)3 ~ a2 \/2 ax - x^ 

/V^^^x = V(a+x)(6+x) 

+ (a- &) log [Va + x + vT+x] 



52 



ENGINEERING MATHEMATICS 



I\h^ ^"^ ^ ^^^ - x) (6 + a;) + {a+b) sin-^ \/^ 
Expressions involving trigonometric forms: 

/sin^ xdx = ~ — -T^milx) 
2 4'' 



J. ^ , sm^-i ^cos:^,/z— 1 C . „ ^ , 
sin"^ ^^t: ax = 1 I sin"*""^ x ax 
w n J 

j cos^ X rfx = ^ + - sin (2 x) 

/cos^ X t/x = - cos"""^ X sin X H / cos**"^ 
n n J 



X ax 



I sin X cos X Jx = I sin^ x 

/ sin^xcos^x Jx = —J [|sin (4x) -- x] 

/cos'^^^ X 
sm X cos"^ X ax = r-r- 
w+ 1 



sin^ X cos X (/x 



sin^+i X 
m+ 1 



cos*^ X sin" X Jx = 



cos"^"^ X sin'^+^ X 



m + n 



fft — 1 1 

H , — I cos'^-^ X sin** X (/x 

m + nj 



cos^xsin'^xt/x = — 



/ 



/ sin^ 
cos"* 



sin"~^ X cos'^^^ X 



+ 



m + n 
n— \ 
m-\- n 



/ 



cos""' X sin'*~2 X dx 



sin^+^ X 



- Jx = > ,. , 

X (fj — 1) cos''~^x 



, n — m — 2 r sin"^ x ^ 
H ^ — I — ;r"9- ^^ 

w — 1 J COS'*"'^ X 



TABLE OF INTEGRALS S3 

/ 'cos^ 
sin'" 



X , _ cos''+^ X 

X {m— V) sin'^"^:^ 



m — n— 2 r cos^ x , 
m— 1 J sin"'~2 X 



m — 2 r dx 



/ dx Q,os>x m— 2 r 

sm^x {m— 1) sin'^~^:\; m— \J \ 



sin'"~2 X 



/dx _ sin X I ^ ~ ^ r ^^ 

cos"" X {n — \) cos"*"^ X n — \ J cos"*"^ x 

I tan xdx = — log cos x 
I tan^ xdx = tan x — x 
j cot xdx = log sin x 
I cot^ :J[; J:;t: = — cot x — x 
/sec X ^:. = log tan (I + I) = i log [±|^ 
I sec^ X J:;t: = tan x 
j cosec :;t: J:v = log tan (J :x:) 

/' 

j x^smxdx = 2xsinx— (x^ — 2) cos a; 

i X cos xdx = cos a: + ^ sin X 

I x^ cos :x: (/x = 2 X cos x + (^^ — 2) sin x 



cosec^ a; fiJx = -- cot x 
sinxdx = siax — X cos r;t: 



m+i 



dx 1 



54 ENGINEERING MATHEMATICS 

Transcendentals 

\ \ogx dx = xlogx — X 

rM^"j.=4.(iog.) 

J X n-\- \ 

/dx , , 

J x([ogxY {n — 1) {[ogxY~^ 

xe'^'^dx = —^{ax— 1) 

a a J 

/'^«^ 1 e«^ g r 6«^ 

j^ ^ ~ ~ m- 1 ^F-i m- 1 J ^^-1 

/. . V , r^ sin (;^x) — n cos (wx)l 
6«^ sm (/zx) J:r = ^«^ ^ — Yjr~2 ^ — 

/. . - Va cos (fix) + ;^ sin (/^:;t:)~| 
6^^ cos (#x) J:^ = 6«^ ^ ^ ' .3 ^-^^ 



HYPERBOLIC FUNCTIONS 
Hyperbolic Transformations 

^x ^ — X 

sinh X = — z = — i sin (/a:) 

where j = V— 1 

cosh :;!£; = = cos [jx) 



dx 



^ 



HYPERBOLIC FUNCTIONS 55 



tanh X = -—-. = — / tan ijx) 

e + e~^ 

coth X = — = / cot ijx) 

e — e 

e^ = cosh X + sinh x 

e~^ = cosh X — sinh X 

sin X = — j sinh {jx) 

cos X = cosh ijx) 



Hyperbolic Formulae 

cosh^ X — sinh^ x = I 

sech^ X + tanh^ x = I 

coth^ X — cosech^ x ^ I 

sinh ix+ y) = sinh x cosh y + cosh ::t: sinh y 

cosh (x + }') = cosh X cosh y + sinh x sinh y 

sinh (x — j) = sinh x cosh y — cosh x sinh 3; 

cosh {x — y) = cosh x cosh y — sinh x sinh y 

, . . _ tanh ::!:: + tanh y 

1 + tanh X tanh y 

, , , . coth X coth 'V + 1 

coth (:x: + ^'j = — -r j — ^-r — 

^ coth 3; + coth X 

, , , V tanh X — tanh v 

tanh {x — y) = 



1 — tanh :^ tanh y 

- , . coth :^ coth a; — 1 

coth {x — y) = — TT ^TT — 

coth y — coth a; 

sinh (2 a:) = 2 sinh x cosh x 

cosh (2 x) = cosh^ x + sinh^ x 

2 tanh x 



tanh (2 x) 



1 + tanh^ X 



56 ENGINEERING MATHEMATICS 

C0th2:r+ 1 



coth (2 x) = 



2 coth X 



(I) = v/' 



. 1 f -^ 1 . / cosh X — 1 
sum 



cosh'"> ..cosh^+1 



(i)=N/: 



tanhi;i-V/':5!^ 

cosh x+ 1 



(0 = V ; 



..... . , cosh X + 1 

coth 



coshx 



sinh X + sinh y = 2 sinh ( — y-^ j cosh f — ^ j 
sinhrJt: — sinhy = 2 cosh I — ^jsinhf "^ j 
cosh :\: + cosh 3; = 2 cosh f — T-^jcosh( — j^j 

cosher — coshy = 2 sinh f — j^j sinh f — j-^j 

sinh (3 :x:) = 3 sinh x + 4 sinh^ x 
cosh (3 :x:) = — 3 cosh x + 4 cosh^ :;»; 

Inverse Hyperbolic Functions 

sinh"^ X = log (x + Vl + x?) 
cosh"^ X = log {x + Va;2 — 1) 

tanh-ix=ilog[[±|] 



HYPERBOLIC FUNCTIONS 57 



sech-i X 


= .o.(i + v/^ 


■) 


cosech"^ : 


«-log(^+v/?- 


^.) 


Differentials of Hyperbolic Functions 


d (sinh x) = 


cosh X dx 




d (cosh x) = 


sinh X dx 




d (tanh x) = 


sech^ X dx 




d{cot\ix) = 


— cosech^ X dx 




d (sech x) = 


— sech X tanh x dx 




d^ (cosech x) 


= — cosech X coth 


:x:^x 


d (sinh~^ x) 


Jx 




Vl + x^ 




d (cosh~^ x) 


dx 
Vx^- 1 




d(t2Lnh-^x) 


1-x^ 




d (coth-i x) 


dx 




d (sech~^ x) 


rfo; 




:^Vl-rt;2 




d (cosech""^ x 


V _ dx 
) / ^ . . 





Use of Hyperbolic Functions 

Illustrative Example. Deduce an expression for 
the length of a perfectly flexible chain suspended be- 
tween two supports; assume that both points of sup- 
port are the same height from the ground. 



58 



ENGINEERING MATHEMATICS 



The chain assumes 
the form of a cate- 
nary (see page 27), 
the equation of which 



IS 



y = a cosh 




The general equation for the length of the chain is 
L = length 



iV'+(g)'- 



ciy 



where the value of -— , obtained by differentiating the 

(JL% 

equation of the catenary, is 



dy 
dx 



I a cosh - j 



dx 



4('^"^a) ©. 



sinh- 
a 



dy . 



Substituting the value of -~ in the formula for the 
length, Lj we have 



L =f\i + sinh2^ dx = Ty cosh2 ^ j^ 

= I cosh - 
J a 



dx = a sinh 



which is the required expression for the length of the 
chain. 

DIFFERENTIAL EQUATIONS 

A differential equation is a relation involving deriv- 
atives or differentials. 
A solution of a differential equation is a relation 



DIFFERENTIAL EQUATIONS 59 

between the variables which satisfies the given equa- 
tion. 

ORDINARY DIFFERENTIAL EQUATIONS 
Equations of the First Order and First Degree 

I. An equation of the form 

fi(x)dx+f2{y)dy = 

can be integrated immediately. 
Its solution is 

J/i (x) dx + J/a {y) dy = C 

An equation may sometimes be changed to the above 
form by separation of the variables. 

II. Homogeneous Equation. An equation is 
homogeneous in respect to its variables when the 
sum of their exponents is the same for each term of the 
equation. 

Homogeneous equations are reduced to the form of 
Method I, by substituting vx for y, and then separating 
the variables. 

III. Non-homogeneous Equation of First Degree 
in X and y. This type occurs in the form: 

{ax + by + c) dx = {a!x + Vy + c') dy 

Substitute for x^ {x! + A), and for y, (y' + k). The 

equation then becomes: 

{ax'-^ by+ah+hk-\-c)dx' = {a'x'+Vy+a'h+b'k+c')dy 

Equating ah+ bk+ c = 

and a'h +b'k + c' =^0 

the original equation now takes the form: 

(ax' + by) dx' = (aV + by) dy 

which is homogeneous and solvable by Method II. 



6o ENGINEERING MATHEMATICS 

In the solution thus obtained, substitute 
x' = X — h and y' = y — k 

where h and k are determined from the two equations: 
ah+bk + c = Q 
a'h + 6'yfe + c' = 

IV. Linear Equation. A Hnear differential equa- 
tion (of first order and first degree) is of the general 
form: 

where P and Q are functions of x alone or constants. 
The solution of this equation is: 

yef'''^=J/'''^Qdx + C 

V. Equations Reducible to the Linear Equation. 

This type occurs in the form: 

where P and Q are functions of x alone. The given 
equation may be written: 

^+(l-n)Pv = (l-n)Q 

where v = 3;"'^+^ This equation is linear in v, and 
solvable by Method IV. In the solution, resubstitute 
for V its value 3;-^+!. 

VI. Exact Differential Equation. An equation of 
the form 

Mdx+ Ndy = 



DIFFERENTIAL EQUATIONS 6 1 

is exact if the derivative of M with regard to y is equal 
to the derivative of N with regard to x. The solution 
then is: 

jMdx+ JIn--^ fudo^dy^C 

where j M dx is the integral of M with respect to x 
(regarding y as constant), and the term 

is found by subtracting from N the derivative in 

respect to 3; of J M dx. The term L^ "~ 7" j M dx\ 

is integrated with regard to y (considering x constant). 
The complete solution is then given by the formula 
above. 

VII. Integrating Factors. If a differential equa- 
tion of the form 

Mdx + Ndy = 

is multipUed through by a certain expression called an 
integrating factor, the equation will become exact. It 
is then solvable by Method VI. 

(a) When an equation is homogeneous, ^ 

is an integrating factor. 

(b) When the condition exists that 

dM _dN 

d^ dx 

= F {x) [an expression containing only x] 

then e^^^^^^^is an integrating factor. 



62 ENGINEERING MATHEMATICS 

(c) Similarly when 

dN^_dM 

^ ^ = F(y) 

then e^^^^^^Hs an integrating factor. 

Equations of the First Order but Higher than 
the First Degree 

In the following formulae, ■— will be denoted by p. 

An equation of first order and of nth degree is of the 
general form 

pn _p jlpn-l J^ ^^n-2 ^ . . . ^ Jp ^ f- = 

where the coefficients A, B, - • - J, K are functions of 
X and y. 

I. Clairaut's Equation. When an equation is of 

the form 

y = px+f(p) 

the solution is obtained by substituting for p a con- 
stant c, 

y = cx+f{c) 

II. Solution by Factoring. The given equation 
may sometimes be resolved into rational factors of the 
first degree. Each factor is equated separately to ^ero, 
and its solution found by one of the preceding methods, 
using the same constant of integration in each case. 
The complete solution is then the product of the sep- 
arate solutions. 

III. Equations Containing only x and p. When 
an equation is of this type, solve for p, and substitute 



DIFFERENTIAL EQUATIONS 63 

its value j- . The resulting equation can be integrated 

immediately. 

IV. Equations Containing only y and p. Solve 

for p, and substitute its value -~, This equation is 

immediately integrable. 

V. Equations Involving x, y, and p. A solution 
can be obtained by one of the following methods: 

(a) Solve for x in terms of y and p. Then differen- 

doc 1 
tiate in respect to y, remembering that — = - . 

ay p 

The solution of this equation, together with the given 

equation, constitutes the complete solution. 

(b) Solve for y in terms of x and p. Differentiate 

dy 
with respect to x, and in place of -p substitute its value 

p. The complete solution consists of the solution of 

this equation, together with the original equation. 

dy 

(c) Solve for ^, and replace it with its value -f^- 

From this equation it may be possible to obtain a 
solution. 

Linear Differential Equations with Constant 
Coefficients 

A linear differential equation is of the first degree 
in the dependent variable and all of its derivatives. 

The particular integral is the solution of the equa- 
tion obtained without the introduction of constants of 
integration. 

The complementary function is the solution ob- 
tained by temporarily equating to zero all those terms 



64 ENGINEERING MATHEMATICS 

of the equation that do not contain the dependent vari- 
able or derivatives thereof. 

The complete solution is the sum of the particular 
integral and the complementary function. 

A linear equation with constant coefficients is of 
the form: 

d^v d^~^v d^~^v 

where the coefficients P, Q, - - • R are constants; and 
X is a function of x. Replacing — by the symbol D, 
the equation becomes 

{D^ + PD^-''+QD^-^+ . . . +R)y = X. 

Case I. Method of Solution when X = o. Write 

the given integral in its symbolic form, replacing -^ 

by D, Then solve this equation for D as if it were an 
ordinary algebraic quantity. 

When the roots of the equation (i.e., the values of D) 
are real, the solution is 

y = cxe^^'' + c^e^"^ + • • • 

where ci, ^2, etc., are the constants of integration, and 
mi, W2, etc., are the roots of the equation. 

When two or more real roots of the equation are 
equal, the solution is 

^^ = (^1 + c^oo + CzX^ + . . . ) e*^* + • • • 

where m is the value of the repeated root, and Ci, ^2, ^3, 
etc., are the constants of integration (introduced in the 
manner shown in the above equation) and equal in 
number to the number of times the root m is 
repeated. 



DIFFERENTIAL EQUATIONS 65 

When the equation has imaginary roots (which 
always occur in pairs) the solution is 

y = e^x^ \A cos {a\x) + B sin {a\x)\ 

+ e^"^^ \C cos (^2^) + Z) sin (02^)] + . . . 

where A and 5, C and Z>, etc., are the constants of 
integration, and \m\ d= a\ V — l), yjfn^ i 02 V— l), etc., 
are the complex imaginary roots of the equation. 

When two or more pairs of complex imaginary roots 
are equal, the solution is 

y = [(^1 + ^2:^ + • • • ) cos {ax) 
+ (^3 + ^4^+ • * • ) sin (ax)] 6"^* 

where (w it a V— l) is the repeated pair of complex 
imaginary roots. 

Case II. Method of solution when X is not 
equal to zero. In this case, the complete solution is 
the sum of the complementary function and the partic- 
ular integral. 

The complementary function is found by tempo- 
rarily equating X = 0, and obtaining the solution by 
the method of Case I. 

The particular integral is obtained as follows. 

The given equation is of the general form: 

(£)- + PZ)^-i + (22^^-2+ • • • ■\-K)y = X 

in which D is used in place of -7- • 
In symboHc notation, this equation may be expressed 

The particular integral can then be written: 
y = Tjjy: = particular integral 



66 ENGINEERING MATHEMATICS 

A. Method of obtaining the particular integral 
when the term X is of the form e^^. 

V pCix pCiX 

particular integral =^_ =^-^ =^^ 

which is found by substituting the constant a in place 
of D. 

This method for evaluating , . fails when the term 

{D — a) is a factor of / {D). The particular integral 
is then found by substituting the constant a for D in all 
terms of / {D) except in the factor {D — a). The 
solution is then completed by the general method given 
under case F (page 68). 

B. Solution for the particular integral when X 
has the form x^. 

particular integral = j^ = 77^ = [f{D)]-^x'^ 

To evaluate this expression, expand [/ {D)]~^ into a 
series of ascending powers of D, by use of the binomial 
theorem. It is only necessary to carry out this expan- 
sion to the mth power of D, since operation on x'^ by 
higher powers of D would produce zero (since the 

symbol D stands for — , the operation by Z> on a 

quantity denotes its derivative with respect to x, the 
operation by D^ denotes its second derivative, etc.). 
In obtaining the solution of the given particular 
integral, x'^ is operated on separately by each term of 
the expansion of [/ {D)]-^, 

C. Method of obtaining the particular integral 
when X has the form sin (ax). 

X sin {ax) 



particular integral = 



/ {D) J (D) 



DIFFERENTIAL EQUATIONS 67 

In order to evaluate this integral, substitute — a^ for 
£)2 wherever U^ occurs in f (D), The particular inte- 
gral will then be a fraction, whose numerator is sin {ax), 
and whose denominator is the value assumed by / {D) 
when D^ is replaced by — a^. 

This method fails if / (D) becomes zero when — a^ 
is substituted for D^, The particular integral is then 
evaluated by writing the term 6'"^ (in which i = V— l) 
in place of sin (ax). The solution of this new integral 
is obtained by method A for the evaluation of the 
particular integral. In the result, ^^''^ is replaced by 
[cos (ax) + i sin (ax)], producing a result containing 
both real and imaginary terms. The required particu- 
lar integral is the coefl&cient of i (i.e., V — l) in this 
expression. 

D. Particular Integral when X = cos (ax). The 
particular integral is obtained as in method C, with the 
exception that cos (ax) is used in place of sin (ax). 

When this method fails, ^'''^ is written in place of 
cos (ax), and this new integral is evaluated by method 
A. In the solution of this integral, e*""^ is replaced by 
[cos (ax) + i sin (ax)]. The required particular inte- 
gral is the real part of this result. 

E. Particular integral when X is of the form e^^Q. 

particular mtegral =^^ =^-^ = '^^fWTaj 

To evaluate the given integral, (D + a) is substituted 
for D, wherever D occurs in / (D) ; and the term e"* is 
treated as a constant multiplier. The new integral 

is evaluated by one of the preceding methods, 



J(D+a) 
or by the general method F. The required particular 



68 ENGINEERING MATHEMATICS 

integral is then equal to the product of ^"^ by the evalua- 
tion of TTTs^-T- 

F. General method for finding the particular 
integral. 

To evaluate jjjyr X 

The denominator of , . may be resolved into factors 

of the first degree. The given integral then becomes: 

1111 1 



{D -a) {D- h) (D -c) {D- d) {D - m) 



X 



The term X is operated on successively by each of these 
fractional operators, beginning at the right. The 

operation on X by the first factor . _ — r produces 

the expression e""^ / ^""^^ X dx. This result is operated 

on in a similar manner by each remaining factor (pro- 
ceeding from right to left). The solution of the given 
particular integral is then: 

Homogeneous Linear Equation 

The homogeneous linear equation is of the form 

in which the coefficients P, ... i? are constants, and 
X is a function of x. 



DIFFERENTIAL EQUATIONS 69 

On assuming the relation, x = e^, this equation may 
be transformed by the substitutions: 

x""^ = d{e-l){e- 2) ' ' ' ton terms 
I ^^^"'£3 = (^- 1) (^- 2) (^- 3) . . . to (^- 1) terms, 
and so forth; where the symbol d stands for — • 

The complementary function is then found as in the 
case of the linear equation with constant coefficients. 
(In obtaining this solution, the term 6 is treated in 
exactly the same manner in which the term D was 
treated in the preceding cases.) 
i In order to obtain the particular integral, the term 

X (which involves only x) is changed to an expression 
involving z, by the substitution x = e^. The particu- 
lar integral is then found by one of the methods given 
under the case of the linear equation with constant 
coefficients. 

The complete solution is the sum of the comple- 
mentary function and the particular integral. In the 
result, z is replaced by its value log x. 

Exact Diflferential Equations 
An exact differential equation is one which can be 
derived directly by differentiation of an equation of the 
next lower order. 
jk If the given equation is of the form: 

where A, B, , . . Q, Rj S, T, and X are functions of 
X, we then have as the condition for exactness that: 



70 ENGINEERING MATHEMATICS 

' j,_dS_^<PR_<PQ^ ... =0 
dx dx^ do(? 

The first integral of the given equation then is: 

dx^-^ "^ V dx Jdx^-^ "^ V ^^ dxydx^-^ 
= Cxdx + C 

This formula may be reapplied successively as long 
as each resulting equation satisfies the condition for 
exactness. 

Equations of the Second Order and the 
First Degree 

General form is 

where P, Q, and X are functions of x, 

I. When one solution of the equation is Imown 
(or can be found by inspection). 

Let yi equal the known integral. In the given 

equation, substitute vyi in place of y; and then, in the 

dv 
transformed equation, replace — by p. This equation 

can be solved by one of the preceding methods. 

II. Change of the Independent Variable. 

The purpose of this change and of the removal of 
the first derivative (see III) is to transform a given 
equation into a new equation which may happen to 
be easily integrable. 

The given equation is of the form: 



DIFFERENTIAL EQUATIONS 7 1 

By changing the independent variable, it may be 
transformed into the following equation: 

where Qi becomes equal to 1, if 





1=^5 


when also 


dh dz 
„ dx''^ dx 

Y ^ 


and 



Q 

or where Pi may be made equal to zero, if 

e"*^ ^dx 



-f 



when also Q\ = -jTr"^ 

dx) 
X 



and Xi = 



\dx) 



III. Removal of the First Derivative. 

To remove the first derivative from an equation of 

the form f| + P^+(3y = Z 

dx^ dx ^^ 

make the substitution y = ve~^'^ ^ 

The given equation then becomes 

where Q,=g-lf_lp3 

and Zi = Ze*-/'^'^ 



72 ENGINEERING MATHEMATICS 

THEORETICAL MECHANICS 

Center of Gravity 

The center of gravity of a body is a point so situ- 
ated that the force of gravity produces no tendency in 
the body to rotate about any axis passing through this 
point. 

Center of Gravity of the Arc of a Plane Curve 

Jxds Jx\/l + (£jdx 

. £* /Vi+(|)V 

where x and y are the coordinates of the center of 

gravity. 

Solve for y in terms of x from the equation of the 

dv 
given curve. Then differentiate in order to obtain -p, 

and substitute its value in the formula for x. 

dx 
Similarly, find x in terms of y, obtain — , and sub- 
stitute in the formula for y. 

Center of Gravity of Plane Areas 
Rectangular Coordinates : 

I J xdA I j xdxdy 



X = 



JJdA ffd.dy 



THEORETICAL MECHANICS 73 

I j ydA I j ydxdy 



y = 



I I dA j j dxdy 



where x and y are the coordinates of the center of 
gravity. 

In evaluating the expression for x, we may integrate 
first either in respect to x or y, according to which 
method is more convenient. 

If dy is integrated first, the Umits of y are expressed 
in terms of x (from the given equation) ; and the Umits 
of X are its initial and final values. 

Similarly, if dx is first integrated, the limits of x are 
expressed in terms of y; and the limits of y are then its 
initial and final values. 

Polar Coordinates : 

/ I r^ cos 6 dddr 
X = ^ ^ 



u 



If 



rdddr 
r^siaddedr 



If' 



rdddr 



Generally, it is more convenient to integrate first 
with respect to r. In this case, the Umits of r are found 
in terms of 6 from the equation of the given curve. 
The Umits of 6 are its initial and final values, expressed 
in radians. 

Center of Gravity of Solids of Revolution. When 
a solid of uniform density is formed by the revolution 
of a plane curve about the X-axis, the center of gravity 



74 ENGINEERING MATHEMATICS 

is on the X-axis (because of symmetry). Its x-coordi- 
nate is 



X = 



I I xydx dy 
j I y dxdy 



where the limits are found as in the case of plane areas. 
When a solid is formed by the revolution of a plane 
figure about the F-axis, the y-coordinate of its center 
of gravity is 

/ j xydx dy 



I I X dxdy 



Center of Gravity of Any Section Composed of 
Two or More Simple Plane Figures 

In order to find the center of gravity of such figures 
as tee-bars, channels, rails, etc., divide them up into 
their component rectangles or triangles. Then, obtain 
the center of gravity and the area of each separate 
figure. Choose any convenient axis in the plane of the 
given section and find the turning moment of each 
figure about this axis. Each turning moment is the 
product of the area of the figure by the distance from 
its center of gravity to the chosen axis. The sum of 
all these separate turning moments gives the turning 
moment of the total figure. On dividing this total 
moment by the total area of the figure, we obtain the 
distance from the chosen axis to the center of gravity 
of the figure. Care must be used, if the chosen axis 
passes through the given figure, to take distances on one 



THEORETICAL MECHANICS 75 

side of this axis as positive, and on the other side as 
negative. 

Generally, one coordinate of the center of gravity 
can be determined by the symmetry of the given sec- 
tion. When the figure is unsymmetrical, it may be 
necessary to take moments about two different axes 
in order to locate the center of gravity. 

Moment of Inertia of Plane Areas 

The moment of inertia of a plane figure about any 
given axis is equal to the integral of the product of each 
elementary area of the figure by the square of its 
distance from the axis. 

Rectangular Moment of Inertia: 

The rectangular moment of inertia of a plane figure 
is its moment of inertia about any axis in the plane of 
the figure. The rectangular moment of inertia of a 
plane area about the X-axis is 



= //: 



y^ dx dy 

The rectangular moment of inertia of a plane area about 
the F-axis is 



I 



= I j x^ dxdy 



In either case, the limits of the variable first integrated 
are expressed in terms of the other variable. 

The moment of inertia of a plane figure about the 
gravity axis (Ig) is its rectangular moment of inertia 
about any axis in the plane of the figure, passing 
through its center of gravity. 

The moment of inertia of a plane figure about any 
axis parallel to the gravity axis and in the plane of 



76 ENGINEERING MATHEMATICS 

the figure is equal to {Ig) plus the product of the area 

of the figure by the square of the distance between the 

two axes, thus: 

I =Ig + Fd? 

Polar Moment of Inertia: 

The polar moment of inertia (Zp) is the moment of 
inertia about any axis perpendicular to the plane of the 
given figure. 

It is equal to the sum of the rectangular moments of 
inertia about two mutually perpendicular axes in the 
plane of the figure, passing through the foot of the 
polar axis. 

In rectangular coordinates, the polar moment of 
inertia equals 

/p = /. + ly =//(^' + f) dxdy 

In polar coordinates, the formula for the polar 
moment of inertia is 



-// 



B?dRde 



It is generally more convenient to integrate first with 
respect to R, expressing its limits in terms of d. The 
limits of 6 are then its initial and final values. 

Moment of Inertia of Solids 

The moment of inertia of a solid (with center at 
origin) about the X-axis is 

I = m j j I (y'^+ z^)dxdydz 

where m is the density, that is, the mass per unit 
volume. 



THEORETICAL MECHANICS 77 

Radius of Gyration 

The center of gyration is that point in a revolving 
body at which, if the entire mass of the body were 
concentrated, the moment of inertia about the axis of 
rotation would be the same as that of the body. 

The radius of gyration, k, is the distance from the 
axis of rotation to the center of gyration. 



For plane sections 



. *=v/-! 



For solids, k.\jL. /X 

in which k = radius of gyration, 

/ = the moment of inertia about the axis of 

rotation, 
A = area of section, 
M = mass of body, 
W = weight of body. 

Center of Percussion 

The center of percussion or oscillation of a pendulum 
or other body vibrating or rotating about a fixed axis 
or center is that point at which, if the entire weight of 
the body were concentrated, the body would continue 
to vibrate in the same intervals of time. 

The radius of oscillation is 

h = 



Md 



iiy 



78 



ENGINEERING MATHEMATICS 



in which I = the moment of inertia of body about axis 
of rotation, 
d = distance from center of gravity of body 

to the axis of rotation, 
h = distance from center of percussion or 
oscillation to the axis of rotation, 
M = mass of body, 
W = weight of body. 



Motion of a Body 

velocity at any instant = v 



ds 

di 



. , . dv d^s 

acceleration at any mstant = ^ = 37 = 



df 



In rectangular coordinates, 

dx _ ds 
dt ~di 



Vx = ^7 = -j: cos d = velocity in a direction parallel to 



the X-axis 



dy ds 



ds /(dxV (dyV 

' = d-r\[dt)+[dt) 

For motion with uniform 
velocity, 



Y 










*/ 




^. 


^ 


— 1 J 



V = J 



For uniformly accelerated motion, 

s = ^ {u + v)t 
s = ut + ^ at^ 
2 as = v^ — u^ 



THEORETICAL MECHANICS 79 

u = initial velocity, 

V = final velocity, 

a = constant acceleration, 

s = space passed over, 

/ = time of motion. 

If the body starts from rest, the initial velocity u 
equals 0, and these equations become: 

s = ^ vt 
s = \ af 

1 as = 1? 



Rotation of a Rigid Body 

velocity at any mstant = ^ = ;77 

acceleration at any instant = a = -rr = "tto 
^ dt dt^ 

For motion with uniform velocity, 

e 

For uniformly accelerated motion, 

^ = i (coo + co) / 

d = angular space through which the body rotates, 
coo = initial angular velocity, 
CO = final angular velocity, 
a = angular acceleration, 
t = time. 



8o ENGINEERING MATHEMATICS 

For a body initially at rest, the velocity coo is 0, 
and these equations become 

e = ^ o)t 
e = ^af 

Falling Bodies 

Equations of motion of a body falling from rest 
under the action of gravity: 

V = gt 

2 gs = v^ 

V = velocity after time /, 

5 — height through which body falls, 

g = (approx.) 32.16 feet/sec.^ = 981 cm/sec.^ 
= acceleration of gravity. 

The value of g for any latitude and any altitude is 

g = 32.0894 (1 + 0.0052375 sin^^) 
X (1- 0.0000000957 £) 
in which 

6 = latitude of place in degrees, 

E = elevation above sea-level in feet. 

Projectiles 

Equations of a body projected vertically upward 
with an initial velocity u (resistance of air not con- 
sidered) : 

(1) Velocity at any time = u — gt, 

(2) Velocity at any height = V w^ __ 2 gh. 

(3) Height at any time = ut^ \ gf. 



THEORETICAL MECHANICS 8l 

(4) Greatest height = — . 

(5) Time of flight = — . 

o 

Equations of a body projected with an initial 
velocity w at an angle 6° to the horizontal (resistance 
of air not considered) : 

The curve described by the projectile is the parabola 
whose equation is 



y = X tan B — 



lu'cos^e 



where B is positive when the body is projected above 
the horizontal and negative when the body is projected 
below the horizontal. 

Horizontal-component of acceleration = "T^i = 

Vertical-component of acceleration = -^^ = ^ g 

(1) Velocity at any time = \^u^ — 2 utg sin B +g^^^. 

(2) Velocity at any height = Vw^ _ 2 gh, 

(3) Height at any time = utsinB — ^ gt^. 

/A\ rr.' r n- t-^ 2wsin^ 

(4) Tune of flight = . 

rc\ -n u^ sin (IB) 

(5) Range = ^ — -. 

o 

If the friction of the air is taken into account, the 

curve described by the projectile is given by the 

empirical relation: 

gx^ / 1 , kx\ 

k = 0.0000000458- 

w 



- I 
82 ENGINEERING MATHEMATICS 

where 

d = diameter of projectile in inches, 
w = weight of projectile in pounds. 

Angular Measure 

A radian is the angle subtended at the center of any 

circle by an arc equal in length to its radius. 

180 
1 radian = degrees = 57.296+ degrees 

TT 

1 degree = 7^7: radians = 0.0175+ radians 
loU 

The relation between the central angle of a circle 
and its subtended arc is given by the formula: 

l = rd 
I = length of arc, 

r = radius of circle, 

6 = central angle in radians. 

Circular Motion 

A body moving with uniform velocity in a circular 
path experiences a constant acceleration toward the 
center of the circle. This acceleration is expended in 
changing the direction of motion of the body. 

The equations of motion of the re- 
volving body are 

a = — 
r 

vT = 2 7rr 

47rV 
^ = -f2- 

V = constant velocity of particle in feet per second, 
a = constant acceleration toward center in feet per 
sec.^, 




THEORETICAL MECHANICS 83 

r = radius of circular path in feet, 
T = time of 1 revolution in seconds, 
7r2 = 9.8696+. 

If the body moves with a variable velocity, then: 
tangential acceleration = -r. 

normal acceleration = — 



Centrifugal Force 

The centrifugal force of a revolving body, in pounds, 
is 

p ^Wf^ ^ 47rWr 

gr ~ g/2 

or in terms of the number of revolutions, iVi, per 
minute 

F = 0.00034 WrNi^ 

W = weight of revolving body in pounds, 
V = velocity of body in feet per second, 
t = time of 1 revolution in seconds, 
r = distance from axis of rotation to the center of 

gravity of the body, in feet, 
g = acceleration of gravity (32.16). 

Flywheel 
The energy of rotation of a flywheel is 

K.E. =^ = IttHN^ 

I = polar moment of inertia about the axis of rotation, 
CO = angular velocity in radians per second, 
N = number of revolutions per second. 



84 ENGINEERING MATHEMATICS 

The energy stored in a rim flywheel by a variation 
in speed is 

W 

E = ~ (S^max — S^min) foot-pOUnds, 

s 
W = weight of flywheel in pounds, 
•Smax = maximum rim speed in feet per second, 
Sndn = minimum rim speed in feet per second 
g = acceleration of gravity (32.16). 

The rim speed in feet per second is 5 = IwRN, 
where N is the speed in revolutions per second, and R 
is the radius of the wheel in feet, measured from the 
center of gravity of the rim section.* 

Hence, the energy stored is 

E = ^ foot-pounds 

and the weight of the flywheel is 

W= ^ 

Substitute for E the required stored energy in foot- 
pounds. Assume some convenient value for R, in feet; 
then solve for the weight W in pounds. If the rim 
speed is too high (average about 35 feet per second for 
cast iron or 150 feet per second for steel), the value of 
R must be reduced. The ratio of the speed variation, 
iVmax — iVmin, to the average speed may be taken as 
follows for different types of machines: 

Hammers 0.20 

Punches 0.05 

Ordinary machinery 0.03 

Textile and paper machinery 0.02 

Electric generators 0.005 

* This value of R is approximately correct. The exact value 
of R is the radius of gyration of the flywheel. 



THEORETICAL MECHANICS 85 

Simple Pendulum 

The time of oscillation in seconds from one extreme 
position to the other is 

^ g 

I = length of pendulum in feet, 

g = acceleration of gravity (32.16 approx.). 

The period of the pendulum is 

^ g 

The seconds-pendulum makes one oscillation per 
second from one extreme position to the other; its 
length in feet is 

Work and Energy 

For a uniform force, 

W 

F = ma = — a 

g 

W 

Ft = mv = ^v 

g 

FS =\mi? = -;y- 
^ g 

F = constant applied force in pounds, 
a = constant acceleration in feet/sec.^, 
m = mass of body, 
W = weight of body in pounds, 

V = velocity acquired after t seconds, 
mv = momentum, 

s = space passed over in feet, 

g = acceleration of gravity (32.16 feet/sec.^). 



86 ENGINEERING MATHEMATICS 

The impulse I of the constant force F during the 
time t equals the change of momentum, 

I = Ft = mv — mu 

where u is the initial velocity and v the final velocity. 
If the force is variable, then impulse equals 



=x 



Fdt 



The work done by a uniform force is 

W = Fs = imv^ 
The work done by a variable force equals 

W=rFds 

The kinetic energy of a body of mass m, moving 
with a velocity v, equals i mv^. 

Direct Central Impact 

For the impact of two bodies of the same material, 
weighing respectively W and Wi pounds, the velocities 
after impact are 

Wu + WiUi — eWi {u — ui) 



V = 



Vi = 



W+Wi 

Wu + WiUi + eW (u — ui) 
W+Wi ' 



u = original velocity of W in feet /second, 
V = velocity of W after impact, 
Ui = original velocity of Wi, 
Vi = velocity of Wi after impact, 
e = coeflQcient of restitution. 



THEORETICAL MECHANICS 



87 



Values of e, the coefficient of restitution, for different 
materials are as follows: 

glass on glass e = .94 

ivory on ivory e = 0.81 

cast iron on cast iron e = 0.66 

lead on lead e = 0.2 

The sum of the momenta of two bodies after impact 
equals the sum of their momenta before impact, 



Wv , WiVi 
g g 



Wu WiUi 



g 



g 



Two inelastic bodies after impact move with a 
common velocity 

_ WiVi + W2V2 
^ " W1 + W2 

in which 

Wi = weight of first body, 
W2 = weight of second body, 

Vi = original velocity of first body, 

V2 = original velocity of second body. 

Composition and Resolution of Forces 

The resultant of the forces 
Fi and F2 acting at a point 
is 



R = VFi^ + 2 F1F2 cos (9 + F2' 

in which d is the angle in 
degrees between the two 
forces. 
The direction of R is determined by the relation 

F2 sin d 




tan a = 



Fi + F2Cosd 



88 



ENGINEERING MATHEMATICS 



in which a is the angle in degrees between Fi 
and R. 

The rectangular components of a force R acting in 
a given direction are 

X = RcosB 
Y = RsinS 

in which X is the horizontal 
component of R, Y is the normal 
component of Rj and d is the angle 
in degrees between R and X, 

The resultant of several forces acting in different 
directions at a point is 




R 



i 



R = VX^+ F2 



in which 



X = Fi cos Oi + f 2 cos $2 

+FzCose,+ . • ' , 
Y = Fi sin 01 + F2 sin 62 

+ F3sin^3+ • •. • , 

where Fi, jF2, F3, etc., are the 

given forces, and ^1, 62, ds, etc., 

are the angles in degrees between the given forces and 

the horizontal axis. 




Friction 



F = friction in pounds, 

N = normal force in pounds, 

/ = coefficient of friction. 

F = fN 
Angle of friction = </> = tan" 


:^ 


■i| = tan-V 



THEORETICAL MECHANICS 



89 



Average values for /, the coefficient of friction, for 
motion are as follows: 



Character of contact 


/ 


Wood on wood 

Metal on wood 

Metal on metal, dry 

Metal on metal, lubricated 

Leather on metal, dry 

Leather on metal, lubricated 


0.25-0.50 
0.50-0.60 
0.15-0.24 

0.075 

0.56 

0.15 



Belt Friction 

P and Q are the forces at the ends of the belt, P being 
the greater force. 

F= resultant force of friction, 
N = normal reaction of pulley, 

6 = angle in radians sub- 
tended by the arc of 
contact, 

/ = coefficient of friction. 

or in common logarithms 
logio^ = 0.434/^ 

The value of / varies from 0.15 to 0.6 depending on 
the condition of belt and pulley, but, in general, it is 
approximately correct to assume/ = 0.3. 

Inclined Plane 

Equations of motion of a body sliding down an 
incline under the action of its own weight. 




90 



ENGINEERING MATHEMATICS 



For a frictionless plane: 



dh 



(1) acceleration along plane = a = — = g sin0, 

dt 

(2) velocity after t seconds = tg sin 6, 

(3) velocity at bottom of plane = ^/l gh, 

(4) distance traveled in t seconds = -^-^ — , 



(5) time of sliding down plane 



=vi 




For an inclined plane with friction 



d^s 



(1) acceleration along plane = ^ = 3^ 

= g [sin^ — fcosd], 
in which 
/ = coefl&cient of friction. 

Conditions for the equilibrium of a body resting on 
an incline: 

W = weight of body, 
F = applied force, 
N = normal pressure on plane, 
6 = inclination of plane in degrees, 
/ = coefl&cient of friction. 





MECHANICS OF MATERIALS 91 

For a frictionless plane: 

(1) When the balancing force is applied parallel to 
the inclined plane, 

F = Wsmd 
N = WcosS 

(2) When the applied force acts horizontally, 

F = IF tan (9, 
N=^W seed. 

For an inclined plane with friction: 

(1) When the balancing force acts parallel to the 
incline, 

^ TF sin (61=1= (90 
cos (^0 

in which 

e' = tan-V 

(2) When the applied force acts horizontally, 

F = WtSin(dzL 6') 

MECHANICS OF MATERIALS 

Stress is distributed force; its intensity per unit 
area is generally expressed in pounds per square inch. 

The elastic limit of a material is the maximum stress 
in pounds per square inch that will be followed by a 
complete recovery of form, after the removal of the 
stress. 

Permanent set is the change in form of a member 
when stressed beyond its elastic limit. 

The ultimate strength of a material is the least stress 
in pounds per square inch that will produce rupture. 



92 ENGINEERING MATHEMATICS 

Modulus of elasticity is the number obtained by 
dividing the actual stress in pounds per square inch by 
the corresponding elongation per inch. 

The factor of safety is the factor obtained by divid- 
ing the ultimate strength by the actual stress in pounds 
per square inch. 

Tension and Compression 
For direct stress, uniformly distributed, 

p = stress in pounds per square inch, 

P = total load in pounds, 

F = cross-sectional area in square inches. 

P 

F_Pl 

I 

E = modulus of elasticity in tension or compression, 
/ = length of member in inches > 
€ = elongation per inch length, 
X = total elongation in inches. 



MECHANICS OF MATERIALS 



93 



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94 



ENGINEERING MATHEMATICS 



Angular Distortion and Shear 
Shearing stress, uniformly distributed equals 



Ps=p 



P = load, 
F = area. 

For torsion: 



E. =^ 



Es = modulus of elasticity in shear, 
b = angle of distortion in radians. 

Note. The modulus of elasticity in shear is f as 
great as in compression or tension. 



b = 



Pa = 



Torsion of Circular Shafts 

Ps = 



eotEs p 



/ 



oilpEi 



ea 
T 

pjp 
e 

^^ 32 

Pa='^ 
^^ 16 



lirPaN 
Horsepower = jj^^qq^^2 

b = helix angle of distortion in radians, 
a = radial angle of distortion in radians, 
/ = length of shaft in inches, 
e = radius of shaft in inches. 




MECHANICS OF MATERIALS 95 

p^ = greatest shearing stress in pounds per square 

inch existing in shaft, 
Eg = modulus of elasticity in shear, 
Ip = polar moment of inertia of circular section (see 

table of standard sections), 
P = force in pounds producing torsion, that is, the 

turning force, 
a = lever arm of force P in inches, 
d = diameter of shaft in inches, 
N = revolutions per minute. 

In deriving the above formulae, the torsion is treated 
as due to a couple of the same turning moment, Pa, as 
the single force P with lever arm a. This eliminates 
the consideration of any stresses other than shearing 
stresses, and, in applying these formulae to the case of 
a single driving force, bending stresses and bearing 
friction are neglected. 

Flexure of Beams 

When a beam is strained by a vertical load, the 
greatest strain will be in the extreme upper and lower 
fibers of the beam. The intensity of the strain that 
can be borne by the extreme fibers is the limit of the 
strength of the beam. The upper fibers are com- 
pressed and the lower fibers are stretched when a 
beam is loaded between supports; the converse holds 
when it is loaded beyond supports. Somewhere along 
or near the center of the beam the fibers are neither 
extended nor compressed; the plane of these fibers is 
called the neutral surface. The line of intersection of 
the neutral surface with any cross-section of the beam 
is the neutral axis of the section. 



96 



ENGINEERING MATHEMATICS 



If the stresses remain within the elastic Umits of the 
material in both tension and compression, and pro- 
vided the modulus of elasticity is the same for both 
kinds of stress, then the neutral axis of the section 
passes through its center of gravity. 

The elastic curve is the curve assumed by a beam 
under load. 

The bending moment for any section of a beam is 
the algebraic sum of the moments of the external or 
appUed forces acting on the beam on one side of the 
section. Thus, for the 
beam shown, the bending 
moment about A is 
M = Rix- Pa 

< The bending moment, 
M, of any section is nu- 
merically equal to the Ri R2 
moment of resistance of 

the section, which is the resistance which the particles 
of the beam offer to distortion. 
The moment of resistance equals 

PL 

e 



1 ' i 


jA 


' ! '^ 



= M = bending moment 



p = stress per unit area at the outermost element of 

the section, 
e = distance of extreme element of beam from neutral 

axis, 
/ = rectangular moment of inertia of beam section 

about its horizontal gravity axis. 

In designing the proper cross-section for a beam, the 
maximum bending moment (given for standard cases 



MECHANICS OF MATERIALS 97 

pi I 

under Beam Loadings) is equated to ^— -. The term -, 

called the section modulus, may be obtained from the 
table of standard sections of beams. The value of p 
must not exceed the maximum allowable stress per 
unit area for the material of the beam. The maximum 
allowable stress equals the ultimate strength divided 
by the factor of safety. 

The equation of the elastic curve and its radius of 
curvature may be found ^rom the relations: 

^, pi EI j^jd^y / X 

ilf =^=--=£/^(approx.) 

E = modulus of elasticity of material of beam in 

tension or compression, 
p = radius of curvature of the elastic curve, 
{x, y) = coordinates of any point on the elastic curve. 

The deflection of a beam at any point is obtained by 
substituting, in the equation of the elastic curve, the 
particular value of x in question, and solving for the 
corresponding value of y, which equals the deflection. 
The maximum deflection occurs at the section for 

which ^ = 0. 
ax 

Shear 

The vertical shear in a beam is equal to the first 
derivative of the bending moment in respect to x, thus 

Vertical shear = J = -^— 

ax 

where M is the bending moment (expressed as a func- 
tion of x). 
The value of the vertical shear for any particular 



98 ENGINEERING MATHEMATICS 

section is found by substituting the corresponding value 
of X in the expression for -7— . The result is the re- 
quired vertical shear. 

The maximum bending moment is found by equat- 
ing -7— = 0, and then solving for the corresponding 

value of X, This particular value of x is substituted in 
the equation of the bending moment,' ikf, and the 
resulting expression equals the maximum bending 
moment. 

The horizontal shear in a plane parallel to the 
neutral surface (that is, the surface in which neither 
tension nor compression occurs), and at a distance z" 
from it, equals 

X (in pounds /sq. inch) = -77^ I zdF 

where / = total vertical shear in pounds, 

y" = width of beam section at z" in inches, 
/ = rectangular moment of inertia of entire sec- 
tion about the horizontal gravity axis, 

Jz dF = area in square inches of that portion of the 

section above 2" multiplied by the dis- 
tance in inches of its center of gravity 
above the neutral axis. 

Beam Loadings 

M = bending moment, 
Mm = maximum bending moment, 
y = deflection at any point, 
d = maximum deflection, 
P = concentrated load, 
W= uniformly distributed load. 



MECHANICS OF MATERIALS 



99 



Cantilever Beam with Concentrated Load 
at the Free End 



M = P{l-x) 

Mn.=Pl 




<A 



^P_ll^_ ^ 



d = 



EI\2 
PP 



3 EI 



m^ 



f 



shear 



moment 



M = 



Cantilever Beam with Uniform Load 

W{1- xf 



21 



Mr 



Wl 







Wx^{ll^+{11-Xf\ R 



l^EIl 



d = 



8 EI 




lOD 



ENGINEERING MATHEMATICS 



Beam Supported at Both Ends and Loaded 
with a Concentrated Load at Center 



M = -7yX 






Px (3/2-4 x^) 



48 £/ 



d=- 



48 EI 



n I 



R 



^ 



R 



T 







shear 


1 



R 



moment 



Beam Supported at Both Ends and 
Uniformly Loaded 



M = 



Wx {I — x) 



21 



y//////^^//////A 



yr _Wl 



y = 



Wx {P -2lx^ + X?) 



24 Ell 



d = 



5WP 
384 EI 




moment^ 



MECHANICS OF MATERIALS 



lOI 



Beam Supported at Both Ends and Loaded 
at Any Point 



i-X-«i 


' 


1 i : 


^ 1 


^■^•^ 


^bJfK^ 


R,l 




shear 


.R2 



M = 




x<a 



,, Pbx „, V 

M = -7 P{x — a) 



x>a 



M, 



Pab 
I 



y^zwi^'^^"-''"-'^ 



x< a 



Pa {I— x) .^ J 2 2\ 

'v = ^ (2 Ix — x^ — a^) 

^ 6 Ell ^ ^ 



x> a 



Pb 



d = ,,^y3{2ab + a^y 



27 Ell 



occurring when x = i v 3 (2 ab + a^) 



I02 



ENGINEERING MATHEMATICS 



I 



Beam Supported at Both Ends and Loaded with 
Two Concentrated Loads at Equal Distances 
from Each End 



1 

i*- a J t a -J 

^ -± 1 



1 , i ' 1 


J 
R 


--^ , T« 






• " n 




E 








shear 




R 






r 


\ 


f 
Mm 


/ 



moment 



M =Px 

M = Pa 

M^ = Pa 

_ Px 

^ Pa 
^ 6£/ 



(2>lx-Zx^- a^) 



'^ 6ElW 



aA 



x<a 
x> a 

x< a 

x> a 



MECHANICS OF MATERIALS 



103 



Beam Fixed at One End, Supported at the Other, 
and with a Concentrated Load at Any Point 



^°4 



W 

Ri 



X'; 



A4^ 



]R, 



shear 




i?i = 



Pa2 (3 / - a) 



2^ 
R2 = P-Ri 

M = P(a-x)-Ri(l-x) 
M = Ri{x-l) 
M„. = Ri{l- a) 
1 



y = 



y = 



6 EI 

1 

6£/ 



(i?i:!c» - 3 i?i?x2 + 3 Pax^ - Po^) 



{Riofi - 3 i?ik2 + 3 Pa^^ - Pa?) 






6 EI 
occurring when 



'-'i^-^i^J 



x< a 
x> a 

x< a 
x> a 



I04 



ENGINEERING MATHEMATICS 



Beam Fixed at One End, Supported at the Other 
and Uniformly Loaded 



y/}iiniiiiiii}^^niimiiK 




"moment 



J?l = 


iw 






i?2 = 


iw 






M = 




4:X){1-X) 




^« = 


Wl 
8 






y = 


Wx" 
AS Ell 


{l-x){Sl- 


2x) 


d = 


■■ 0.0054 







EI 



occurring when x = 0.5785/ 



MECHANICS OF MATERIALS 



105 



Beam Fixed at Both Ends and Loaded at the 

Center 



i : 1 



^ 










shear 


1 



Mmj\^ 



ft 
Mm 



flioment 



M = |(4x-0 



M™ = — 



8 






<Z = 



48 £/ 
192 EI 



io6 



ENGINEERING MATHEMATICS 



Beam Fixed at Both Ends and Uniformly Loaded 




w 



y = 



wi 

12 

24 Ell 

WP 
384 EI 



Q - xy 



COLUMNS 

Note. The breaking load in Euler's and in Gordon's 
formula, and the safe load in Ritter's formula are in 
pounds. In all of the formulae for columns, the length, 
I, and radius of gyration, k, must be expressed in the 
same units (generally inches). 

Euler's Formula 

(1) Column with round ends, 

breaking load = El'^=ir^EF (pj 



tm 



MECHANICS OF MATERIALS 107 

(2) Column with flat ends, 

breaking load = 4£/^ = ^ir^EF (^j 

(3) Pin-and-square column (column with one end 
round and the other flat), 

breaking load = -EI j^ = -w^EF Ij^j 

in which 

E = modulus of elasticity of material of column in 

tension or compression, 
/ = rectangular moment of inertia of cross-section 

about neutral axis, 
I = length of column, 
F = area of cross-section in sq. inches, 
k = least radius of gyration of section. 

Gordon's or Rankine's Formula 

(1) Column with flat ends, 

breaking load = - 



2 

1+ "■" 



HI) 

(2) Colunrn with rounded ends, 

FC 

breaking load = rrrg 

(3) Pin-and-square column, 

FC 

breaking load = rrr^ 



io8 



ENGINEERING MATHEMATICS 



in which 

F = area of cross-section in square inches, 

C = ultimate compressive strength of material of 

column in pounds per square inch, 
I = length of column, 
k = least radius of gyration of section, 
^ = empirical constant. 

Values of /5 and of C, in Gordon's formula, are as 
follows for different materials: 



Material 


Hard 
steel 


Medium 
steel 


Soft 
steel 


Wrought 
iron 


Cast 
iron 


Timber 


C (lbs./ 
sq. in.). 


70,000 


50,000 


45,000 


36,000 


70,000 


7200 


^ 


1 


1 


1 


1 


1 
6400 


1 




25,000 


36,000 36,000 


36,000 


3000 



Ritter's Formula 

(1) Column with flat ends, 

FC 



safe load = 



1 + 



c /iv 



(2) Column with rounded ends, 

FC 



safe load = 



i + ^C-Y 



(3) Pin-and-square column, 



safe load = 



FC 



1 + 



1.78 C /IV 



47r2£ \k 



MECHANICS OF MATERIALS IO9 

in which 

F = area of cross-section in square inches, 

C = maximum safe compressive stress of material of 

column in pounds per square inch, 
C = compressive stress at elastic limit in pounds per 

square inch, 
E = modulus of elasticity for tension or compression, 
I = length of column, 
k = least radius of gyration. 

J. B. Johnson's Formula 

Breaking load in pounds; cross-section in square 
inches. 
For mild steel: 

(1) Pin-ends, 

breaking load = [42,000 - 0.97 {t)^F 



7 ) not > ISO 



(2) Flat ends, 

^ = I 4-9 nnn — n <S9 / 



breaking load = [42,000 - 0.62 f^V \F 




(e 



not > 190 



For wrought iron: 

(1) Pin-ends, 

breaking load = [34,000 - 0.67 (i)'\f 

I 



, ,not>170 



no ENGINEERING MATHEMATICS 

(2) Flat ends, 
breaking load = [34,000 - 0.43 (r)^^ 

(|)not>210 
Notation same as in Ritter's formula. 

Straight-line Formula 

Breaking load in pounds; cross-section in square 
inches. 

For mild steel : 

(1) Hinged ends, 

breaking load = [52,000 - 220 (^\]f 

(2) Flat ends, 

breaking load = [52,000 - 179 (^'If 

For wrought iron : 

(1) Hinged ends, 

breaking load = [42,000 - 157 (~\ F 

(2) Flat ends, 

breaking load = [42,000 - 128 (^ F 

Notation same as in Ritter's formula. 

Wooden Columns 

The breaking load in pounds for solid wooden col- 
umns with square ends is 

(700+ 15 m)FC 



P = 



700+l5m + m^ 



MECHANICS OF MATERIALS 



III 



F — cross-section in square inches, 
m — ratio of the length, /, of the column to the least di- 
mension (/, of the cross-section ( that is, w = ^ j, 

C — ultimate compressive strength of material of 
column in pounds per square inch. 

Values of C, the ultimate compressive strength, for 
different kinds of timber are as follows: 

White oak and Georgia yellow pine 5000 Ib./sq. in. 

Douglas fir and short-leaf yellow pine . . . 4500 Ib./sq. in. 

Red pine, spruce, hemlock, cypress, chest- 
nut, CaUfornia redwood, and Cali- 
fornia spruce 4000 Ib./sq. in. 

White pine and cedar 3500 Ib./sq. in. 

The proper factor of safety for yellow pine varies 
from 3.5 to 5, according to the amount of moisture 
present in the timber, being greater for larger amounts 
of moisture. For all other timbers, the proper factor 
of safety varies from 4 to 5. 



CENTERS OF GRAVITY 
Plane Figures 

Triangle 
The C.G. is on a median line of T 
the triangle, two-thirds of its length j^ 
from the vertex. 



^ = 3 




Parallelogram 
The C.G. is at the inter- 
section of the diagonals, 
- h 

y-2 




112 



ENGINEERING MATHEMATICS 



y = 



Trapezoid 

3 (a + 6) 




Quadrant of Circle 
_ 4r _ 
^ = 31^ = ^ 



r = 



4 rV2 




Quadrant of Ellipse 

_ 4a 

^-. 4& 
^ = 37 




Semicircle 
- 4r 
^ = 3^ 




4 r 



Circular Sector 

_ 2 re 



y = 



3a 




MECHANICS OF MATERIALS 



113 



Circular Segment 

- — 4 r sin^ d 
^~ 3 2 (9 - sin (2 Q) 

is in radians 




Sector of a Circular Ring 



X — IZ 



2R^ -r^sind 



SR'-r^ d 




Parabolic Segment 

_ 3h 




Parabolic Segment 

x = ib 




ir I 



h. 



114 



ENGINEERING MATHEMATICS 



Solids 



Hemisphere 




Right Pj^amid or Cone 

- h 



y-i 




MECHANICS OF MATERIALS 



IIS 



MOMENT OF INERTIA OF SOLIDS 

W 
M=mass of body= — 



Shape of figure 



^^ 



r2r^ 



^2R 





Descrip- 
tion 



uniform 
thin rod 



thin rec- 
tangular 
plate 



thin 

circular 

plate 



solid 
cylinder, 
radius, r 



hollow- 
cylinder, 
i?= outer 

radius, 
r= inner 

radius 



solid 

sphere, 

r= radius 



hollow 
sphere, 
jR= exter- 
nal ra- 
dius 
r= inter- 
nal ra- 
dius 



Axis of rotation 



(1) through center 

perpendicular 
to length 

(2) through end per- 

pendicular to 
length 



(1) through center 

of gravity per- 
pendicular to 
plate 

(2) through center 

of gravity paral- 
lel to side b 



(1) through center 

perpendicular to 
plane 

(2) any diameter 



(1) axis of cylinder 

(2) through center of 

gravity, perpen- 
dicular to axis 
of cylinder 



(1) axis of cylinder 

(2) through center 
of gravity per- 
pendicular to 
axis 



through 
center 



through 
center 



Moment of 
inertia 



/2 

M- 



M 



M 



M 



12 
12 









M 



(^i) 



M 



'R2+y 



"(n* 



.) 

^2+^ 



M 



2r^ 



M 



2 / J?5-y5 X 

5 \R^-r^) 



ii6 



ENGINEERING MATHEMATICS 





>> 


1 




s 


fe 






r> 


o 






+j 


0) 


c 


-t-> 


<u 


a 


B 


?^ 


o 
S 




^ 


o 






^ 


K.^ 



|vO 







^H 






rO 






+ 






W 












•^ 


-'^ 








^ 


•-^ 


+ 


^C> 


n^ 


fri 


1 




f^ 






>^^ 


•^ 




►«|(N 




/"^ 


»<:>l^ 




fC^ 



C^ 






'%^ II 

(DO «o 



S.1VO 



Sb 



11^ 



V) 

rt Q 



C^ <4-l 

^ O 
CO 



^N 



S.I2 ^12 



CN 



^12 






^IS SP 



CN 



CM 



\t 



<< M 



<5 



►55 



I 



^0 



^0 



11^ 



■^^ 



.C^ 






K^ 



^ 



m 



MECHANICS OF MATERIALS 



117 



I 



CO 



+ 



Nl'* 



I 



-Td 









•^ 






'^ 



I 
CO 






vO 



^12 





% 


OS 


s 



N 



-IS 



I 



^0 1^ 



O 



10 

o 



^ 






SH 



s^ 








VO 


00 




rCi 


vO 


(N 


1 


00 


00 


1 


. 






•^ 










►o 




Vii 



t<-j=-^ 



'Ff 



ii8 



ENGINEERING MATHEMATICS 






o 
\—{ 

O 

P 

< 
m 

O 

»— I 
P^ 

Ph 

o 






ss 



-IS 



+ 

eo 



-12 



r^ 



eo 



I 











^^1 




0) 


T3 ^' 6 


^ 
•«-> 


d 

II 

1^ _ 
d 'd 


rmine 
page? 

tie mo 


^ 
.S 


dete 
n on 
s. T 




^11 




+^ W> £>. 


^ 


O 5i 


canno 
ethod 
gravit 




o ^ 


1 


l-o 


y of which 
is by the m 
s along the 


1 

o 


11 

o7> 


ravit 
ty aJi 
base 


1 

Xf\ 


en tj 
d t 


of g] 
gravi 
their 


-o 


^•§ 


t 


6 ::; 


center 
izontal 
s, with 




|1 


^1^ 


■o 1 
II 


?. o 


ngle-bars, 
ion of the 
nent recta 


3 


.52 § 


cC .ti O 


<i> 


^11 


•4J 




rt ., o 


>. 


earns 
nd the 
its c 




al 


^ 


i2 g 


asT-b 
irst, fi 
ion int 


3 




2 


«« «r 


sections 
oUows: F 
e the sect 


CO 


d 2i 


i2 


.2 S) 


ia of such 
ained as f 
hen divid 


-3) 




inert 

obt 

. T 


d 


£1 

O M 


<n xn 




rj 


oment of 
, may b 
e Section 






S o-^ 


_c3 


3 'a 


(U -f-J o 


Ui 


o ^? 


sx ^. ^ 


(U 


■^ QJ 


H ^ a 


Pi 


<u-ti 






^ 


.So 


O 


H ;=! 



HYDRAULICS II9 

HYDRAULICS 
Head and Pressure 

The difference in level of water between two points 
is called the head. 

The pressure in pounds per square inch at any depth 
is 

p = 0.433 h 

in which 

h = head or depth in feet of water, 
0.433 = weight of a column of water 1 foot high and 
1 inch in cross-section. 

The pressure on a submerged surface is always 
normal to the surface, and equals 

P (in pounds) = 0.433 hF 

h = depth of water in feet from the surface of the 
liquid to the center of gravity of the sub- 
merged surface, 

F = area of submerged surface in square inches. 

Center of Pressure 

The center of pressure of a submerged surface is the 
point of application of the resultant of all the fluid 
pressures on such surface. 

The distance of the center of pressure of a vertical 
submerged plate below the liquid surface is 

d (in feet) = ~ 



120 ENGINEERING MATHEMATICS 

F = area of plate in square feet, 
i = distance in feet from the liquid surface to the 

center of gravity of the plate, 
la = rectangular moment of inertia of plate about 

the line of intersection of its plane with the 

surface of the liquid. 

The distance of the center of pressure of a sub- 
merged plate inclined at an angle d with the surface is 

d (m feet) = -^-^ h z 

rz 

z = distance from the liquid surface to the center 

of gravity of the plate in feet, 
F = area of plate in square feet, 
Ig = moment of inertia of plate about its gravity axis 
parallel to the liquid surface. 

Flow through Apertures 

Due to friction, the velocity of discharge through an 
aperture in a thin plate or plank is reduced about 3 
per cent below its theoretical value. Further, on 
leaving the orifice, the jet contracts to approximately 
64 per cent of the area of the aperture. 

The theoretical velocity of discharge through a small 
aperture, in feet per second, is 

^ = VYgh 
g = acceleration of gravity = 32.16, 
h = head in feet. 

The actual velocity of discharge in feet per second is 

V = (l> VJgh = 0.97 Vigh 
(j) = coefficient of velocity. 



HYDRAULICS 121 

The discharge through the aperture in cubic feet per 
second is 

Q = CF<j>\/Ygh = 0,62 pVlJh 

C = 0.64 (approx.) = coeflftcient of contraction, 
F = area of aperture in square feet. 

FLOW OF WATER IN PIPES 

Bernoulli's Theorem 

A general method for calculating the flow of water in 
pipes is given by Bernoulli's theorem* 

that is, the sum of the velocity head ;r— , the pressure 

^ S 

p 
head - and the potential head z at any given section 

of flow is equal to the sum of the corresponding heads 
at any other section, plus the various losses between 
the two sections considered. 




V = velocity in feet per second at first section, 
I'l = velocity at second section, 

p = pressure in pounds per square inch at first 

section, 
pi = pressure at second section, 



122 ENGINEERING MATHEMATICS 

z = potential head at first section in feet, that is, 
the distance of the center of the section 
above a chosen horizontal reference plane, 

Zi = potential head at second section, 

g = 32.16 (approx.), 

7 = weight in pounds of a column of water 1 foot 
high and 1 square inch in cross-section = 
0.433, 

k = various losses in feet of head between the two 
sections of pipe considered. 

Losses in Pipes 

The following formulae for losses in pipes enable us 
to find the value of the term k appearing in Bernoulli's 
theorem. If several losses occur in a section of pipe, 
the total loss, kj is the sum of the separate losses. 

Loss Due to Friction 

The loss of head in feet due to friction in a sec- 
tion of pipe is 

where 

d = diameter of pipe in feet, 
I = length of pipe in feet, 
V = velocity in feet per second, 
/ = coefficient of friction, depending on the velocity, 
and on the size of pipe. 

Values of /, the coefficient of friction, for water in 
clean iron pipes are as follows (condensed from I. P. 
Church's '^Mechanics of Engineering "): 



HYDRAULICS 



123 



Veloc- 


Diam. 

1 :„ 


Diam. 

1 ir-i 


Diam. 


Diam. 


Diam. 


Diam. 


Diam. 


Diam. 


ity in 


-2 in. 




= 2 in. 


= 4 in. 


= 8 in. 


= 12 in. 


= 16 in. 


= 20 in. 


feet per 
second 


0.0417 

ft. 


0.0834 

ft. 


= 0.1667 

ft. 


= 0.333 

ft. 


=0.667 

ft. 


= 1.00 

ft. 


= 1.333 

ft. 


= 1.667 

ft. 


0.1 


0.0150 


0.0119 


0.00870 


0.00763 


0.00704 


0.00669 


0.00623 




0.3 


0.0137 


0.0113 


0.00850 


0.00750 


0.00693 


0.00657 


0.00614 


0.00578 


0.6 


0.0124 


0.0104 


0.00822 


0.00732 


0.00677 


0.00642 


0.00603 


0.00567 


1.0 


0.0110 


0.00950 


0.00790 


0.00712 


0.00659 


0.00624 


0.00588 


0.00555 


2.0 


0.00862 


0.00810 


0.00731 


0.00678 


0.00624 


0.00593 


0.00559 


0.00529 


3.0 


0.00753 


0.00734 


0.00692 


0.00650 


0.00600 


0.00570 


0.00538 


0.00509 


6.0 


0.00689 


0.00670 


0.00640 


0.00605 


0.00562 


0.00534 


00507 


0.00482 


12.0 


0.0%30 


0.00614 


0.00590 


0.00560 


0.00522 


00500 


0.00478 


0.00457 


20.0 


0.00615 


0.00598 


0.00579 


0.00549 


0.00508 


0.00485 








Loss at Entrance 

The loss of head in feet due to entrance from a 
reservoir into a pipe is equal to 

in which is the co- \ zJyz:~Ir~~ ^ ^ - 

eflScient of friction and \—_ ^ -=-> ~ ~~~^ ~ - — 

is dependent on the \r__rr~_~'/ 

angle d° which the pipe ^" """ " ^ 

makes with the inner surface of the reservoir. 

Values oi Le[ = —^— 1 ) in the above formula are as 

follows for different values of 6° (from Church) : 



^° 


90° 


80° 


70° 


60° 


50° 


40° 


30° 


Le 


0.505 


0.565 


0.635 


0.713 


0.794 


0.870 


0.987 



Thus, when the discharge is through a pipe normal to 
the inner surface of the reservoir, then ^° equals 90"^ 
and Le is, therefore, 0.505, the loss at entrance then being 

c,2 



0.505 



2g 



where v = velocity of flow in pipe in feet per second. 



124 



ENGINEERING MATHEMATICS 



Loss Due to Sudden Enlargement 

The loss of head in feet due to the sudden enlarge- 
ment of a pipe is 

.2 



(^'J| 



g 



Fi = cross-section area of ! ~ 

the smaller pipe in ; 

square feet, ^ i 

F = area of enlarged ^ 

section in square feet, 
V = velocity in feet per second in the enlarged 

section. 



2g 



Loss Due to Sudden Contraction 

The loss of head in feet due to the sudden contraction 
of a pipe is 

in which 

V = velocity in feet per 

second in con- "" 
tracted section, 
C = coefficient of con- 
traction, the 
value of which 
depends on the 

F 

ratio, — , of the small section to the large 

section. 
Values of C, the coefficient of contraction, for 



F 



HYDRAULICS 



125 



diflferent values of — are given in the following table 
(from Church): 



F_ 



0.10 



0.624 



0.20 



0.632 



0.30 



0.643 



0.40 



0.659 



0.50 



0.681 



0.60 



0.712 



0.70 



0.755 



0.80 



0.813 



0.90 



0.892 



1.0 
1.0 



Loss Due to Bends 

The loss of head in feet due to a bend in a circular 
pipe is 



[o- 



131 + 1.847 






2g 



a = radius of pipe in feet, 
r = radius of bend in feet, 
V = velocity of flow in feet per second. 

Values of Lb for different values of - are as follows: 

r 



U 



0.10 



0.131 



0.20 



0.138 



0.30 



0.158 



0.40 



0.206 



0.50 



0.294 



0.60 



0.440 



0.70 



0.661 



0.80 



0.977 



0.90 



1.40 



1.00 



1.98 



Flow Through Straight Cylindrical Pipes 

Q = discharge in cubic feet per second, 
V = velocity of discharge in feet per second, 
I = length of pipe in feet, 
d = diameter of pipe in feet, 
Le = coefficient of loss at entrance. In general, the 

pipe is normal to the inner surface of the 

reservoir and then Le = 0.505. For other 

cases see Loss at Entrance. 
/ = coefficient of friction, obtained from the table 

on page 123. 



126 ENGINEERING MATHEMATICS 

(1) Required the head in feet necessary to keep up 
a given flow of Q cubic feet per second in a clean iron 
pipe of given length / and diameter d. 

The required head is 

/i (in feet) = |^(l + Le + 4/^) 
4Q 



in which v = 



7rd^ 



(2) Required the velocity in the pipe, having given 
the head h and the length I and the diameter d of the 
pipe; also required the discharge Q in cubic feet per 
second. 

The velocity in feet per second is: 



.= ■ '^' 



l + Le+4/^ 



and after solving for Vy 

Q = lirdh 

Since the value of / depends on the unknown v as 
well as the known J, we may first put / = 0.006 for 
a trial approximation and solve for v; then take the 
value of/ corresponding to this velocity and substitute 
again in the given formula for v. One trial is generally 
sufficient for ordinary accuracy. 

(3) Required the proper diameter d for the pipe to 
discharge a given quantity Q cubic feet per second, 
having given the length of pipe and the head h. 

The proper diameter in feet is 



^'^'-"^W^c?)' 



HYDRAULICS 127 

and d being solved for, 

Since the radical contains d, we must first assume a 
trial value for d, and taking / = 0.006, substitute in 
the above formula for the diameter. Having obtained 
a value for d, we solve for the velocity v. With the 
approximate values of d and v thus obtained, we find 
the corresponding new value of / from the table of 
friction, and then substitute again in the formulae. 
One or two trials generally give sufficient accuracy. 

Flow Through Very Long Pipes 

When a pipe is very long (1000 feet or more), the 
head, velocity, or discharge, etc., may be calculated 
from the formulae: 

I 1? 
/f = 4/- r— (Chezy's formula) 
dig. 



40 



Notation same as in preceding section. 



FLOW THROUGH OPEN CHANNELS 
Bazin's Formula 
The velocity of flow in a channel in feet per second is 



V = 



0.552 + -^ 

Vr 



28 ENGINEERING MATHEMATICS 

r = mean hydraulic radius in feet, which is fou'nd 
by dividing the area of the fluid cross-section 
in square feet by the wetted perimeter in feet 
(that is, the perimeter of the channel section 
in contact with the water), 

s = slope of stream (that is, the difference in eleva- 
tion between two points of the water surface 
divided by the distance between the two 
points measured along the surface), 

m = coefficient of roughness, the values of which are 
given in the following table. 



Character of channel 



Very smooth cement surfaces or planed boards. . 

Concrete, well-laid brick, unplaned boards 

Ashlar, good rubble masonry, poor brickwork. . . 

Earth beds in perfect condition 

Earth beds in ordinary condition 

Earth beds in bad condition covered with debris 



Value of m 



0.06 
0.16 
0.46 
0.85 
1.30 
1.75 



Kutter's Formula 

The velocity of flow in a channel in feet per second 

equals 

,, ^^ , 0.00281 , 1.811 
41.65 H 1 

_ ,s n /— 

where r and 5 are as in Bazin's formula. 

Values for n^ the coefficient of roughness, are as 
follows: 



HYDRAULICS 



129 



Character of channel 



Planed timber, glazed or enameled surfaces. . 

Smooth clean cement 

Unplaned timber, new well-laid brickwork. . . 
Smooth stonework, ordinary brickwork, iron. 

Rough ashlar and good rubble masonry 

Firm gravel 

Earth in ordinary condition 

Earth with stones, weeds, etc 

Earth or gravel in bad condition 



Value of n 


0.009 


0.010 


0.012 


0.013 


0.017 


0.020 


0.025 


0.030 


0.035 



FLOW OVER WEIRS 

Contraction is complete when no edge of the weir is 
flush with the sides or bottom of the channel. 

Contraction is incomplete when one or more sides 
of the weir have an interior border flush with the sides 
or bottom of the channel. 

Francis' Formula 
The flow over a weir in cubic feet per second is 
(2 = 1 [0.622 h{h- tV nh) VYgh] 
in which 

h = head in feet of water on weir, 
b = width of weir in feet, 
n = 2 for complete contraction, 
n = I ior one end of weir flush with side of channel, 
n = for both ends of weir flush with sides of 
channel. 

Bazin's Formula for Weirs 

For overfall- weirs with end contractions suppressed, 
the flow in cubic feet per second is 



130 ENGINEERING MATHEMATICS 

in which the coefficient n has the value 

0.0148 



n = 0.6075 + 



h 



h = depth in feet of water on weir, 
b = width of weir in feet, 

p = height in feet of the sill of the weir above the 
bottom of the channel of approach. 

STRESSES IN PIPES AND CYLINDERS 

Pressure in Pipes 

The tensile stress in pounds per square inch in a pipe 
due to internal fluid pressure is: 

fp 

For thin pipes, P' = ^ 

t 

For thick pipes or cylinders, 

p(r+t) 

r = inside radius of pipe in inches, 
t = thickness of pipe in inches, 
p = excess of internal over external pressure in 
pounds per square inch. 

If S is the required factor of safety, then: 
For thin pipes, t = S-^ 

For thick pipes or cylinders, 

rp 



t 



P- pS 
in which r and p are as above, and 

P = ultimate tensile strength of material of pipe (see 
Table of Strength of Materials). 



FLOW OF FLUIDS I31 

Collapsing of Tubes 

The collapsing pressure for Bessemer steel lap- 
welded tubes, for lengths greater than six diameters, is 

p =1000 (1 - y 1 - I6O0Q when ^ < 0.023 

or 

p = 86670 3 - 1386 when -. > 0.023 

a a 

(Stewart's equations) 

in which 

p = excess of external over internal pressure in 

pounds per square inch, 
d = outside diameter of tube in inches, 
/ = thickness of tube wall in inches. 

FLOW OF FLUIDS 

Flow of Air Through Apertures 

The weight of air in pounds discharged per second 
from a reservoir into the atmosphere is 



M=0.^3F-|L when pi>2pa 

or . 

M = 1.06 F J tAhlJ:A when pi<2pa 



Fliegner's 
equations 



pi = reservoir pressure in pounds per square inch 

absolute, 
pa = atmospheric pressure (14.7 pounds per square 

inch) , 
F = cross-section of aperture in square inches, 
T\ = absolute temperature of reservoir (degrees Fahr. 

+ 459-6). 



132 ENGINEERING MATHEMATICS 

Flow of Steam Through Apertures 

M = 0.0165 Fpi'-^' (Grashof's formula) 

M = ^ when pi> ^p2 



1^ Pp2. I^ipl- P2) T_ . ^ 5 



Napier's 
equations 



Grashof's formula applies when the final pressure is 
less than 58 per cent of the reservoir pressure. 

M = pounds of steam discharged per second, 
pi = reservoir pressure in pounds per square inch, 
p2 = final pressure in pounds per square inch, 
F = cross-section of aperture in square inches. 

Flow of Gas in Pipes 

Q = 1000 y^ (Molesworth) 

Q = quantity of gas in cubic feet per hour, 

d = diameter of pipe in inches, 

/ = length of pipe in yards, 

h = pressure in inches of water, 

5 = specific gravity of gas relative to air. 

Flow of Air in Pipes 

^ = 114.5 y^ (Hawksley) 

V = velocity in feet per second, 
h = head in inches of water, 
d = diameter of pipe in inches, 



FLOW OF FLUIDS 133 

L = length of pipe in feet, 

Q = quantity in cubic feet per second. 

Flow of Compressed Air in Pipes 



i.0.U61v/^'. 0.1161 v^^' 

Q = volume in cubic feet per minute of compressed 

air, at 62° F., 
Qi = volume before compression, at 62° F., 
r = pressure in atmospheres, 
p = difference in pressures in pounds per sq. inch, 

causing the flow, 
d = diameter of pipe in inches, 
L = length of pipe in feet. 



Flow of Steam in Pipes 






(Babcock) 



W = weight of steam flowing in pounds per minute, 
w = density in pounds per cubic foot of the steam 

at the entrance to the pipe, 
pi = pressure in pounds per square inch at the 

entrance, 
p2 = pressure at exit, 
d = diameter in inches, 
L = length of pipe in feet. 



134 



ENGINEERING MATHEMATICS 



ELECTRICITY 

OHMIC RESISTANCE 

The resistance of a uniform electric conductor at 
0° Centigrade is given by the formula: 

R (in ohms) = p-j 

L = length of conductor in inches, 
A = cross-section in square inches, 
p = resistivity of conductor at 0° C, values of which 
are given in the following table. 



TABLE OF RESISTIVITIES 

(Resistivity is the resistance in ohms between any two 
opposite faces of a 1 inch cube of the material)* 



Metal 


Resistivity at 
0°C. 


Aluminium (annealed) . . 
Aluminium (commercial) 

Aluminium bronze 

Bismuth (compressed) . . . 
Brass 


1.14 XlO-^ 
1.05 XlO-6 
4.96 XlO-^ 

51.2 XlO-6 
2.82 XlO-^ 
0.637x10-6 
0.625x10-6 
8.23 XlO-6 
0.803x10-6 
3.82 XlO-6 
7.68 XlO-6 
1.72 XlO-6 

37.1 XlO-6 
4.89 XlO-6 
3.53 XlO-6 
0.575x10-6 
5.16 XlO-6 
2. X10-« 
2.28 XlO-6 


Copper (drawn) 

Copper (annealed) 

German silver . . . 


Gold (annealed) 

Iron (wrought) 


Lead (compressed) 

Magnesium 


Mercury 


Nickel (annealed) 

Platinum (annealed) 

Silver (annealed) 

Tin 


Tungsten 


Zinc (pressed) 





* This definition applies to English units and to the numerical values 
given in the table. In general, resistivity is the resistance of a unit cube. 



IS 



ELECTRICITY 135 

The resistance of a conductor at any temperature 

(1 + ah) 

in which 

Ri = known resistance at a temperature h degrees 

Centigrade, 
R<2, = required resistance at a temperature h degrees 

Centigrade, 
a = temperature coefficient of electrical resistance, 
the value of which is given for different metals 
in the following table. 



TEMPERATURE COEFFICIENTS OF ELECTRI- 
CAL RESISTANCE 



Metal 


Temp, coeffi- 
cient (approx.) 
forl°C. 


Aluminium (commercial) . . 

Copper (annealed) 

German silver 


0.00435 
0.00388 
0.00036 
0.00365 
0.00463 
0.00072 
0.00247 
0.00377 
0.00570 
0.00365 


Gold (annealed) 


Iron (wrought) 


Mercury 

Platinum 

Silver 

Tungsten 


Zinc (pressed) 



Note. — The temperature coefficient of a material is its increase in resistance 
for each degree Centigrade rise in temperature, and it is expressed as a decimal 
fraction of the resistance at 0° C. 



136 



ENGINEERING MATHEMATICS 



u 

Qi .J 

o — 



moo >0'>«'0^ 






0^\00 0^r<^0 
o tr»fso i>.sO»n 
f<^<N<N — .— »— 



O (D 



(DO 
00 
C — 



o: 



r^scsi— sovo-'r <NfNO\ msoo\ r^ 

•^r^os »— '^oo cr^o^^o vooom <^oooo t^vOi>t tncTiso 

000 ,— — — rsi<Nr«^ -^uM^ o^'— -^ oon^os r>.t>»a» 

odo 000 000 000 o — — ^ — <N<Ni fA'^'m' 







0<^0<> OsOON «— f«^0^ 

VO'^f^ o<Niri om — 

000 — — — PvICNfO 



c<^00— 00 
OO-^ 0<NOO 



000 000 000 000 O — — ^<N<N 



o 

C/3 



0^r^— f^OOO vO-^O <^ — -"r — 0<N vTiOvO 

\0<Nir» <Nin(N — <NvO OsOco ooovo in'«rf<^ <n<n — 

\0«^0 oo>om -"rr^fN rq — — — oi~ ~ 

— — — 000 000 000 OOi 



000 000 000 000 000 



o o o o o o 



:8: 



00c 

ooc 
o_^o^c_ 



>OQ 000 000 000 000 
>oo 000 OQr^ f<^00— ifiooiO 
ir>i— c<>oom — •^<N "^— :;— *^'0.^ 



n 

s 






















q 
'3)0 
















— 00 vol^ro ^oov oot^i>* vomin 


m 


1 











8 


s 


r< 


» 


S?§ ^§5: sPiS c^;^^ 






O — (N rO'^vn \OI>«00 O^O— CNfA^ lAsOr^ 



ELECTRICITY 



137 



Is 

O — 
Ah 



a 

o 

.s 



f^ 



O- 



&^ 



oo^ o-'J"'* ■* — — <NOin mor>» r^o^ 

fsoov m-Tf-^t- <Nt^so — Qooo o-fo tncNo t>»^^ 52'^ 

o^o^o "^o^io cNo^r^ vO'^f<^ <^rvj— — — o oco oo 

3 000 000 000 00 



CN — — —00 



T^o^ —00 CN — O vO-^cr* 






173 r>N 

oil 

O 



vO 00 o 



— tnoO CNOvO >A<N'^ 



— vom fOfNf*^ 000 
<N<Nr<^ -"rinvo 000 



03 



00 — 

<NOQO 

sii 



moo —mo mfs 

vOm-rr rO<N«N — — 

§88 888 88 

000 000 000 



000 000 000 000 




■^«N00 sOvOr<^ 0000 

ON — ■«»■ OMncN o^rs. 

c<^f<^r^^ _,— — 00 

o 000 00 

- 000 00 

888 88 

000 00 



000 

<NOv<N 
vO«NO 



2i;!8 

oovcm -n-focN cs — — 



(NOI^ 

Tfo otNin ovOtN 
scm -nr 






2. ^ 
P5 



00f<^0 



lAOvO 


— OOs 


<NvOcO 


OC^O^ 


_c<^so 











<^ c*^ c<^ 


^iQ« 


?:|^Si 


00 NO in 


lATfcn 


CM 









c 
o 

O <u 

|8 



inovo 


— 0>0n 


(NvOcn 


OONO 


— <r\vo 


in 








f<^«S<N 


;5;?52: 


oosom 


•<^<N — 


— oo^ 


00 









in (^ vo — o^On p^ vO r<^ 



oo^o 
1 0000 



— mso 
t>ivOin 



in — 
men 









ooo^o — p^m '♦•invo r>.ooa^ o — <n pf^'^m 
— — <N <N«Nfs <NPvjp^ <N<NCN m co m cn m m 



OnO 



138 ENGINEERING MATHEMATICS 

Ohm's Law 

or E = IR 

I = current in amperes, 

E = electromotive force in volts, 

R = resistance in ohms. 

The proper size of wire in circular mils for any direct 
current circuit on a two-wire system consisting of 
copper conductors is given by the formula: 

10.8 X 2dXl 
cm. = — -g - 

or if the resistance is required, 

E 
r = 



2dXl 
where 

r = resistance per foot of wire in ohms, 
E = volts drop in line, 
/ = total line current in amperes, 
d = distance from source to load in feet, 
cm. = cross-section of conductor in circular mils. 

Resistance of Circuits 

The resultant of several resistances in series equals 
R = ri + r2 + r3+ • • • 

where ri, ^2, ^3, etc., are the separate resistances. 

The resultant of several resistances in parallel or 
multiple is given by the relation: w 

• 1+1+1+.. . 

R n r2 Tz 



ELECTRICITY 139 

R is the total or combined resistance; and ri, ^2, rs, etc., 
are the separate resistances. 

Power and Energy in Direct Current Circuits 

The power in watts expended in a resistance is 

P = EI = PR 

E = electromotive force in volts, 
/ = current in amperes, 
R = resistance in ohms. 

The energy transformed into heat in a time t seconds 

is 

e = EIt= PRt 

when the current, /, is constant; or, if the current is 
variable, energy equals 



!> 



i^Rdt 

h 

where i is the instantaneous value of the current, 
expressed as a function of t. 

The power in any two-wire direct current circuit is 

P (in watts) = EI 

where E is the volts between the terminals of the cir- 
cuit and / is the current in amperes. 

MOTORS AND GENERATORS 

The frequency in cycles per second is given by the 
relation: 

_ R.P.M. P 
•^ " 60 ^2 

R.P.M. = speed in revolutions per minute, 
P = number of poles. 



I40 ENGINEERING MATHEMATICS 

Equations of Direct Current Motor 
The armature current of a motor, during starting, is 

Ra H~ Rx 

in which 

E = impressed voltage, 

e = counter-electromotive force, 
Ra = armature resistance in ohms, 
Rx — resistance of grid or rheostat in series with 
armature. 

At full speed, 

E- e 
^"" Ra 
e = K(j>f 

E =IaJla+e=IaRa+K<t>f 

E - K(j>f 






Ra 
E — IgRa 

K<t> 



f = frequency in cycles per second,* 
</) = total field flux in magnetic lines, cutting arma- 
ture conductors, 
K = constant for any given machine. Its value is 

4/ 

77^> where t is the number of armature turns 

in series. 

* Frequency, in the case of a direct current machine, refers to 
the frequency of alternation in the armature windings, not, of 
course, in the external circuit* 



ELECTRICITY 141 

Equations of Direct Current Generator 

E = e- laRa 
e = generated voltage, 
E = terminal voltage, 
la = armature current in amperes, 
Ra = armature resistance in ohms. 

R = resistance of load in ohms. 

E^RIa 
e=E+IaJla=Ia(R+Ra) 

Torque 

The torque of a dynamo in foot-pounds equals 

T = KI<I> 
where 

<^ = total field flux in magnetic lines, cutting arma- 
ture conductors, 
/ = armature current in amperes, 
K = constant term for any given dynamo. Its value 

2.348 
is K = ' Q tP, t being the number of arma- 
ture turns in series, and P the total number 
of poles. 

The torque of a motor in terms of the horsepower is 
^^ 33,000 H.P. 

or solving for horsepower, 

lirTn IwRFn 



H.P. = 



33,000 33,000 



142 ENGINEERING MATHEMATICS 

n = number of revolutions per minute, 
T = torque in foot-pounds, 
R = radius of pulley in feet, 
F = turning force in pounds. 

Induced Voltage 

N dci> . 

' = ^WTt ^^^^' 

N = number of turns. 

If the turns cut across a uniform field, at right angles 

to the lines of force, then -j- equals the number of lines 

cut per second. Otherwise, — is the first derivative 

of ^ in respect to t, (j> being expressed as a function of /. 

The efifective voltage induced in the windings of a 

generator, motor, or transformer, etc., is given by 

the relation: 

^ V2Tfn(t> 4.44/^0 ,, 
£ = -I^=--li- volts 

This formula is generally quite accurate, being derived 
on the assumption of uniform flux distribution. 
/ = frequency in cycles per second, 
(l> = total number of lines of magnetic force, 
n = effective number of turns. If all the turns are 
grouped in one coil, then n equals the total 
number of turns. Otherwise, if the winding 
is distributed over k electrical degrees (as in 
the armature of a motor or generator), then 



the effective number of turns isn = N 



k 
2 

N being the total number of turns. 



i 



ELECTRICITY 143 

The average induced voltage of a dynamo is 
E - 4^ volts 

where n is the number of armature turns in series. 



Inductance 

Inductance, L, is the number of interHnkages of flux 
with turns, per unit current, 

L (henrys) = j^ 

in which 

N = number of turns, 
/ = current in amperes, 

(j> = number of Hues of magnetic force interhnking 
with the turns. 

The theoretical unit of inductance is the centimeter. 

The practical unit of inductance is the henry, which 
equals 10^ centimeters. 

The counter-electromotive force in an inductive 
circuit is 

^ di 

provided the inductance, L, is constant. 

The total voltage consumed by an inductive circuit 

E.ir+L% 

the inductance, L, being constant. 

di . 
r is the resistance of the circuit in ohms, and -r- is the 

first derivative of i with respect to ^, the current i being 
expressed as a function of t. 



144 



ENGINEERING MATHEMATICS 



The inductance in henrys of an air-core circular 
coil is 

0.366 ■ 



L = 



Viooo/ 



F' 
F 



106+ 12c + 2Jg 
106+ 10c+ lAR 

UR 



XF'F" 



" = 0.51ogio( 



100 + 



2 & + 3 c/ 
I = length of conductor in feet. 



— 




- -T 1 


- 




f a 

r I 1 




• b 1- 




t 

1 







All other dimensions are in inches and as indicated in 
the diagram. 

The inductance, L, of a concentric cable in henrys 
per 1000 feet is 

L - 105 X 

j^+4.61og.o^+^^^,_^,^, 



4.6 J?o^ ^0 



l 3Ro^- R^ l 



where 

r = radius of inner metallic conductor, 
R = distance from center of cable to the inner sur- 
face of the outer metallic conductor, 
Ro= distance from center of cable to the outer sur- 
face of the outer metallic conductor. 



ELECTRICITY 145 

The values of r, R, and Rq must be expressed in the same 
units. 

The total inductance, L, of a two- wire transmis- 
sion circuit in henry s per 1000 feet is 

^ 3.048 (^^ , D-r , 

where 
/zi = permeabihty of the metal conductor; for copper, 

Ml = 1, 
IX = permeability of medium separating wires; for 

air, jjL= ly 
D = distance between the two lines, measured from 

center to center, 
r = radius of conductor, in same unit as D. 

Capacity 

The unit of capacity is the farad. Since the farad 
is very large, the microfarad, which is one-millionth of 
a farad, is used as the practical unit. The theoretical 
unit of capacity is the centimeter, 9 X 10^^ centimeters 
being equal to 1 farad. 

The charge of a condenser, Q, is measured in ampere- 
seconds or coulombs, and may be calculated by the 
formula: 

Q = CE 



from which 


^=1 


and 
where 




C = capacity in farads, 

E = potential across the terminals of the condenser 
m volts. 



146 



ENGINEERING MATHEMATICS 



The capacity of a plate condenser is 
22AS KA 



C = 



microfarads 



I 



d X W 
where 
A = total area in square inches of all the dielectric 

sheets separating the condenser plates, 
d = average thickness in inches of one sheet of the 

dielectric, 
K = inductivity of the dielectric, average values of 
which are given in the following table for 
different materials. 



Materials 


Induc- 
tivity K 


Air (at standard pressure). 
Manilla paper 


1.00 
1.50 
2.00 
2.50 
2.50 
3.00 
3.00 
3.10 
6.00 


Paraffin, solid 


Ebonite 


India rubber 


SheUac 


Oil 


Glass 


Mica 





Condensers in Parallel. When two or more con- 
densers are connected in parallel, the resultant capacity, 
C, equals the sum of the separate capacities, thus 

C=Ci+C2 + C3+ .... 

Condensers in Series. When two or more con- 
densers of capacities Ci, C2, C3, etc., are connected in 
series, the resultant capacity is given by the formula: 

1 



C = 



-^ + ^- + ^ + 



ELECTRICITY 147 

The capacity, C, of a concentric cable per 1000 feet 
in microfarads is 

7.37 



C = 



lOOOlogio- 
P 



in which 

p = radius of inner metallic conductor, 
Po = distance from center of cable to the inner sur- 
face of the outer metallic conductor, in the 
same unit as p. 

The capacity, C, of a two-wire transmission line 

per 1000 feet in microfarads is given approximately by 

the formula: 

3.68 



C = 



lOOOlogio^ 



i 



r 

if the lines are not close to the ground. 

D = distance between the two wires of the trans- 
mission line, measured from center to center, 
r = radius of conductor, in same unit as D. 

The differential equations of a condenser are 

dg = idt 

q = charge = j idt 

dq = cde 
de 

Alternating Current Circuits 

The shape of the voltage or current wave produced 
by an alternator is, in general, nearly that of a sine 
curve. Alternating current calculations are, therefore, 
usually worked out on this assumption. 




148 ENGINEERING MATHEMATICS 

The number of cycles or complete waves per second 
is the frequency of the current, and the time required 
for the current to 
complete one cycle 
is a period. 

The average 
value of the current 
or voltage is the 
average of all the 
ordinates of the curve of one half- wave. The effective 
value of an alternating current or voltage is the square 
root of the sum of the squares of the instantaneous 
values of a half-wave. 

If E is the maximum voltage of a half-cycle of a sine 
wave, 

2 
average voltage =-£ = 0.636^ 

TT 
1 

eflfective voltage = —rzE = 0.707 E 

Similarly, if the maximum current is /, 

2 
average current = -/ = 0.636/ 

TT 
1 

effective current = 77^/ = 0.707/ 

When the voltage reaches a definite value in the cycle 
sooner than the current reaches its corresponding value, 
the voltage and current are out of phase with each 
other; the voltage is said to be leading, and the current 
to be lagging. Phase difference is always expressed in 
degrees; a complete cycle equals 360 degrees. 



ELECTRICITY 149 

Alternating Voltage and Current 

or E = IZ 

I = current in amperes, 
E = electromotive force in volts, 
Z = impedance in ohms. 

Impedance and Reactance 

r = resistance in ohms 
X = reactance in ohms 
z = impedance in ohms 

The relation between resistance, reactance, and im- 
pedance is the same as that between the three sides of 
a right triangle. 

r = z cos a 

X = zsina 

a = tan ^- 
r 

z = VrM^ 

Inductive Circuits 
The inductive reactance in ohms is 
Xl = 27r/L 

where / = frequency in cycles per second, 
L = inductance in henrys. 

The impedance in ohms is 

z = VrM^ = V/'2+47r2/2L2 




150 ENGINEERING MATHEMATICS 

Circuits having Capacity 
The capacity reactance in ohms is 

1 

ItJC 

where / = frequency in cycles per second, 
C = capacity in farads. 

The impedance in ohms is 



Circuits having Inductance and Capacity 
The reactance in ohms is 

X = xl + xc= 2 tt/L - j^ 

The impedance in ohms equals 

z = Vr^+{xL + xcY 

Vector Representation of Sine Waves 

A sine wave of voltage or current may be represented 
by a vector, the magnitude or length of which is equal 
to the effective value of the sine wave. It is some- 
times more convenient to let the length of the vector 
equal the maximum value of the sine wave. The 
vector is generally denoted by a capital letter, with a 
dot directly beneath it; it is expressed in terms of its 
rectangular components, which determine the magni- 
tude of the vector and its direction relative to the 
coordinate axes. Thus, the vector E is written 

E = e + je' 
in which j = V— 1 

where e denotes the horizontal or real component of the 



ELECTRICITY 



151 



vector, and e' the vertical or imaginary component. 
The imaginary unit, 7, in the above equation, merely 
denotes the direction of measurement of e\ 




The magnitude of E is 

E = Ve^ + e'2 

and the angle B which the vector E makes with the 
horizontal axis is 



d 



tan-i - 



The angle in degrees between two vectors is the 
phase difference between the two sine waves which the 
vectors represent. 

In vector notation, the 
impedance is 

Z =^ r -\- ]x 
and its magnitude is 
Z = Vr2 + x^ 
The admittance is 

Z r -{- jx 




r . X , 



where g = ^ = conductance, 



* = -| 



susceptance. 



152 



The 



ENGINEERING MATHEMATICS 

current 



equals 



E 



r = ~ = EY = {e+je') 






+ ji' 



and the voltage is 

E = IZ = (i+ji') {r +jx) = e +jV 



Power in Alternating Current Circuits 

If the effective voltage and current are represented 
by the vectors 

E = e+jV 



E 









A 




T 








/ 


^^ 


• 




^ 


y^6 


^ 


e' 


• f 






— e 


"T 


k 



the real power is 

IF = a + eV = EI cose 

the wattless power is 

Wi = e'i- ei' = EI sinff 

the volt-amperes equals EI, 

The power-factor is the cosine of the angle between 
the voltage and current vectors, 

power-factor = cos ^ = — ^ — 



ELECTRICITY 



153 



Balanced Three-phase Circuits 

E = volts between lines 
e = volts per phase 
/ = current in each line 
i = current in each phase 

Line 



Phase 




Y connection 



For Y-connections, 



E = e Vs ; e = —7= ; and I =i 

V3 



Line 



;Phase 




^ - connection 

For A-connections, 

£ = ^; / = i V3; and i = —tz^ 

V3 

In either case, for non-inductive load, the power in 
watts is 

W = V3EI 



154 ENGINEERING MATHEMATICS 

If the load is inductive, then the power is 
W = V3 EI cosd 
where cos 6 is the power-factor of the phase. 

MAGNETISM 

Equations of Magnetic Circuits 

F = attractive or repellent force in dynes, 
mmf = magnetomotive force in ampere turns, 
N = number of turns, 

/ = current in amperes, 

jS = density in magnetic lines per square centi- 
meter, 

<j> = total number of lines of flux, 
A = cross-section of magnetic path in square 
centimeters, 

fjL = permeability, 
H = intensity of field, 

I = length of magnetic circuit in centimeters, 

p = reluctance, 
m = pole strength, 

r = distance between poles. 

QAttNT 



V ■ 

p = 

4> : 


0.4 


P 
ttNIhA 




I 


^■ 


. 
A 




mmf 


0.4 

= 0.4 


tNIh 
I 

irNI 


M 


H 





MAGNETISM 



ISS 



Magnets and Magnetic Fields 
F = mH 



I 



^ mm 

(f) = Arirm 

The attractive force in pounds exerted by a two 

pole magnet is P = y ^^^ , where 5 is the total 

area of both pole faces in square inches, and B is the 
density in magnetic lines per square inch. 

The ampere-turns required to maintain a flux den- 
sity of B lines per square inch in an air gap is IN = 
0.313 Blj in which I is the length of the gap in inches. 

Hysteresis Loss 
The power in watts lost in hysteresis is 

, fVB'-' 



W 



107 



/ = frequency in cycles per second, 
V = volume of iron in cubic inches, 
B = magnetic density in lines per square inch, 
k = empirical constant, values of which are given 
in the following table. 



Character of iron 


Value of k 


Silicon steel 


(J. 0006 to 0.00075 
0.0008 to 0.0011 
0.010 to 0.012 
0.013 to 0.017 


Annealed sheet iron. . . . 
Cast steel. . . 


Cast iron 



156 ENGINEERING MATHEMATICS 

Eddy Current Loss 

The power in watts lost due to eddy currents in iron 
or steel laminations is approximately 

/ = frequency in cycles per second, 
I = average thickness of lamination in inches, 
B = magnetic density in lines per square inch, 
V = volume of iron in cubic inches. 

This formula holds for ordinary temperatures, and if 
the thickness of the lamination is not greater than 
0.025 inch. In silicon steel, the eddy current loss is 
approximately \ of that given above. 

STANDARD SATURATION CURVES 

B = density in lines per square inch 
A T/in. = ampere-turns per inch 

Values of ampere-turns per inch for densities not included 
in the following tables may be determined approximately 
by interpolation. Thus, the AT/in. for silicon steel for 
B/sq.in. = 65,500 is 

AT/in, = 4.5 + ^^ (6.4 - 4.5) = 5.5 (approx.) 



MAGNETISM 



157 



Silicon Steel 


Annealed Sheet Iron 


Saturation curve 


Saturation curve 


B 
30,000 
40,000 
50,000 
60,000 
70,000 
80,000 
90,000 
100,000 
110,000 


AT /in. 

2.1 
2.7 
3.4 
4.5 
6.4 
10 
23 
35 
100 
225 
520 
1000 
2200 
3770 
5330 
6900 


B 
30,000 
40,000 
50,000 
60,000 
70,000 
80,000 
90,000 
100,000 


AT /in. 

4 

4.4 

5 

9 
12 
20 
33 
60 


120,000 




130,000 






135,000 






140,000 






145,000 






150,000 






155,000 














Cast Steel 


Cast Iron 


Saturation curve 


Saturation curve 


B 

50,000 

60,000 

70,000 

I 80,000 

f 90,000 

100,000 

105,000 


AT /in. 
11 

15 

20 

29.5 

50 
105 
165 


B 

5,000 
10,000 
15,000 
20,000 
25,000 
30,000 
35,000 
40,000 
45,000 
50,000 
55,000 


AT /in. 
8 
12 
17 
23 
30 
43 
60 
85 






110 






145 






190 









158 ENGINEERING MATHEMATICS 

MEASUREMENT 
English Weights and Measures 

Length 

1000 mils = 1 inch , 
12 inches = 1 foot 

3 feet = 1 yard 
5280 feet = 1 mile 

4 inches = 1 hand 

9 inches = 1 span 
2 J feet = 1 pace 

16| feet or 5^ yards = 1 rod 
1 knot or nautical mile = 6080.26 feet 
= I league 

7.92 inches = 1 link 
25 links = 1 rod 
100 links or 66 feet or 4 rods = 1 chain 

10 chains = 1 furlong 

8 furlongs = 1 mile 

Surface 

144 square inches = 1 square foot 

9 square feet = 1 square yard 
30i square yards = 1 square rod 
160 square rods = 1 acre 

640 acres = 1 square mile 

625 square links = 1 square rod 
16 square rods = 1 square chain 

10 square chains = 1 acre 

640 acres = 1 square mile 

36 square miles = 1 township 

Volume 

1728 cubic inches = 1 cubic foot 

27 cubic feet = 1 cubic yard 

128 cubic feet = 1 cord 

24f cubic feet = 1 perch 



MEASUREMENT 159 

Troy Weight 

24 grains (gr.) = 1 pennyweight (dwt.) 
20 pennyweights = 1 ounce (oz.) 

12 ounces = 1 pound (lb.) 

Avoirdupois Weight 

16 drams (dr.) = 1 ounce (oz.) 
16 ounces = 1 pound (lb.) 

25 pounds = 1 quarter (qr.) 

4 quarters = 1 hundred weight (cwt.) 

20 hundred weight (2000 pounds) 
= 1 ton (T.) 

Apothecaries' Weight 

20 grains (gr.) = 1 scruple (sc. or 9) 

3 scruples = 1 dram (dr. or 3) 
8 drams = 1 ounce (oz. or §) 
12 ounces = 1 pound (lb) 

Dry Measure 

2 pints (pt.) = 1 quart (qt.) 
8 quarts = 1 peck (pk.) 

4 pecks = 1 bushel (bu.) 
36 bushels = 1 chaldron (ch.) 

Liquid Measure 

4 gills (gi.) = 1 pint (pt.) 
2 pints = 1 quart (qt.) 

4 quarts = 1 gallon (gal.) 
31i gallons = 1 barrel (bar.) 
63 gallons = 1 hogshead (hhd.) 

Apothecaries' Fluid Measure 

60 minims = 1 fluid-drachm 

8 fluid-drachms = 1 fluid-ounce 
16 fluid-ounces = 1 pint 
8 pints = 1 gallon 

Circular Measure 

60 seconds (") = 1 minute (') 

60 minutes = 1 degree (°) 

30 degrees = 1 sign (s) 

12 signs, or 360 degrees = 1 circle (cir.) 



i6o 



ENGINEERING MATHEMATICS 



English and Metric Conversion Factors 



1 millimeter 
1 centimeter 
1 inch 
1 foot 
1 yard 
1 meter 

1 kilometer 

1 mile 



Length 

= 39.37 mils 

= 0.03937 inch 

= 0.3937 inch 

= 0.0328 foot 

= 2.54 centimeters 

= 0.083 foot 

= 30.48 centimeters 

= 0.305 meter 

= 91.44 centimeters 

= 0.914 meter 

= 39.37 inches 

= 3.28 feet 

= 1.094 yards 

= 3280.8 feet 

= 1093.6 yards 

= 0.6214 mile 

= 5280 feet 

= 1609.3 meters 

= 1.609 kilometer 



1 circular mil 
1 square mil 

1 sq. millimeter 

1 sq. centimeter 
1 sq. inch 
1 sq. foot 



Surface 

= 0.7854 square mil 

= 0.0005067 square millimeter 

= 1.273 circular mils 

= 0.000645 square miUimeter 

= 0.000001 square inch 

= 1973 circular mils 

= 1550 sq. mils 

= 0.00155 sq. inch 

= 197,300 circular mils 

= 0.155 sq. inch 

= 1,273,240 circular mils 

= 6.4516 sq. centimeters 

= 929.03 sq. centimeters 

= 144 sq. inches 



MEASUREMENT 



i6i 



1 sq. yard 

1 sq. meter 

1 are 
1 acre 



1 hectare 

1 sq. kilometer 

1 sq. mile 



1296 sq. inches 
9 sq. feet 
0.00836 are 
0.000207 acre 
1550 sq. inches 
10.76 sq. feet 
1.196 sq. yards 
1076 sq. feet 
100 sq. meters 
43,560 sq. feet 
4840 sq. yards 
4047 sq. meters 
0.4047 hectare 
0.001562 sq. mile 
107,600 sq. feet 
100 ares 
2.471 acres 
10,764,111 sq. feet 
247 acres 
0.3861 sq. mile 
27,878,400 sq. feet 
640 acres 
2.59 sq. kilometers 



Volume 

1 cu. centimeter = 0.061 cu. inch 

= 0.0021 pint (liquid) 
= 0.0018 pint (dry) 

1 cu. inch = 16.39 cu. centimeters 

= 0.0173 quart (hquid) 
= 0.01488 quart (dry) 
= 0.0164 liter or cu. decimeter 
= 0.004329 gaUon 
= 0.0005787 cu. foot 

1 quart (liquid) = 2 pints (liquid) 

= 946.36 cu. centimeters 

= 57.75 cu. inches 

= 0.94636 liter or cu. decimeter 

1 quart (dry) = 2 pints (dry) 



l62 



ENGINEERING MATHEMATICS 



1 quart (dry) 



1 liter 



1 gallon 



= 1101 cu. centimeters 

= 67.20 cu. inches 

= 0.03889 cu. foot 

= 1000 cu. centimeters 

= 61.023 cu. inches 

= 1.0567 quarts (dry) 

= 0.2642 gallon 

= 3785 cu. centimeters 

= 231 cu. inches 

= 3.785 liters 

= 0.1337 cu. foot 






Note. Pints, quarts, and gallons in this table refer to U.S. measures. 



1 milligram 
1 grain 
1 gram 

1 ounce (av.) 

1 poimd (av.) 

1 kilogram 
1 ton (short) 

1 ton (metric) 



Weight 

= 0.01543 grain 

= 0.001 gram 

= 64.80 milligrams 

- 0.002286 ounce (av.) 

= 15.43 grains 

= 0.03527 ounce (av.) 

= 0.002205 pound (av.) 

= 437.5 grains 

= 28.35 grams 

= 0.0625 pound (av.) 

= 7000 grains 

= 453.6 grams 

= 16 ounces 

= 0.4536 kilogram 

= 35.27 ounces 

= 2.205 pounds 

= 2000 pounds (av.) 

= 907.2 kilograms 

= 0.8928 ton (long) 

= 0.9072 ton (metric) 

= 2205 pounds 

= 1000 kilograms 

= 1.102 ton (short) • 

= 0.9842 ton (long) 



MEASUREMENT 163 



1 ton (long) = 2240 pounds 

= 1.12 ton (short) 
= 1.016 ton (metric) 



Force 

1 dyne = 0.01574 grain 
= 0.00102 gram 
= 0.00007233 poundal 
= 0.000002248 pound (av.) 

1 gram = 980.6 dynes 

= 0.07093 poundal 

1 poimdal = 13,825 dynes 
= 0.03108 pound 
= 0.01410 kilogram 

1 pound = 444,800 dynes 
= 32.17 poundals 

1 kilogram = 980600 dynes 
= 70.93 poundals 



Storage of Water 

1 acre-foot = 325,800 gallons 
= 43,560 cu. feet 
= 1613 cu. yards 
= 1233 cu. meters 
1 gallon = 0.000003069 acre-foot 
1 cu. foot = 0.00002298 acre-foot 
1 cu. yard = 0.00062 acre-foot 



Temperature 

1 degree Centigrade = | (= 1.8) degree Fahrenheit 

1 degree Fahrenheit = f ( = 0.556) degree Centigrade 

temperature Fahr. = // = | /c + 32 

temperature Cent. = ^c = I (^/ — 32) 



164 



ENGINEERING MATHEMATICS 



1 gram-centimeter 



1 joule 



1 foot-pound 



Heat, Electric, and Mechanical Equivalents 

Energy 
1 erg = 1 dyne-cm. 

= 0.0000001 joule 

= 0.00000007376 foot-pound 

= 980.6 ergs 

= 0.00009806 joule 

= 0.00007233 foot-pound 

= 10,000,000 ergs 

= 0.7376 foot-pound 

= 0.2389 gram-calorie 

= 0.102 kilogram-meter 

= 0.0009480 B.t.u. 

= 0.0002778 watt-hour 

= 13,560,000 ergs 

= 1.356 joules 

= 0.3239 gram-calorie 

= 0.1383 kilogram-meter 

= 0.001285 B.t.u. 

= 0.0003766 watt-hour 

= 0.0000005051 horsepower-hour 

= 9.806 joules 

= 7.233 foot-pounds 

= 0.009296 B.t.u. 

= 0.002724 watt-hour 

= 1055 joules 

= 778.1 foot-pounds 

= 252 gram-calories 

= 107.6 kilogram-meters 

= 0.2930 watt-hour 

= 0.0003930 horsepower-hour 

= 3600 joules 

= 2655.4 foot-pounds 

= 860 gram-calories 

= 3.413 B.t.u. 

= 0.001341 horsepower-hour 

= 4186 joules 

= 3088 foot-pounds 



1 kilogram-meter 



1 B.t.u. 



1 watt-hour 



1 kilogram-calorie 



MEASUREMENT 165 

1 kilogram-calorie = 426.9 kilogram-meters 

= 1.163 watt-hours 
1 horsepower-hour = 2,684,000 joules 

= 1,980,000 foot-pounds 

= 745.6 watt-hours 

Power 

1 erg per second = 1 dyne-centimeter per second 

= 0.0000001 watt 

1 gram-centimeter per second = 0.00009806 watt 

1 foot-pound per minute = 0.02260 watt 

= 0.00003072 horsepower (metric) 
= 0.00003030 horsepower 

1 watt = 44.26 foot-pounds per minute 

= 6.1 19 kilogram-meters per minute 

1 horsepower = 33,000 foot-pounds per minute 

= 745.6 watts 

= 550 foot-pounds per second 
= 1.01387 horsepowei (metric) 

1 horsepower (metric) = 32,550 foot-pounds per minute 

= 735.5 watts 

= 75 kilogram-meters per second 
= 0.9863 horsepower 

1 kilowatt = 44,256.7 foot-pounds per minute 

= 1.3597 horsepower (metric) 
= 1.341 horsepower 

Electric Units 

1 abvolt = 10-8 volt 
1 abampere = 10 amperes 
1 abohm = 10-® ohm 

Pressure Eqiiivalents 

B 1 atmosphere (standard) = 29.92 12 inches of mercury at 32° F. 

= 760 miUime ters of mercury at 32 ° F. 
= 33.901 feet of water at 39.1° F. 
= 14.6969 pounds per sq. inch 
= 2116.35 pounds per sq. foot ( 



l66 ENGINEERING MATHEMATICS 

1 inch of mercury at 32° F. = 0.491187 pound per sq. inch 

= 70.7310 pounds per sq. foot 
= 1.13299 feet of water at 39.1° F. 

1 foot of water at 39.1° F. = 0.8826 inch of mercury at 32° F. 

= 62.425 pounds per sq. foot 
= 0.4335 pound per sq. inch 
= 0.0295 atmosphere 

1 pound on the sq. foot = 0.016018 foot of water at 39.1° F. 

1 pound on the sq. inch = 2.307 feet of water at 39.1° F. 

PRESSURE AND VOLUME CORRECTION, ETC. 
Reduction of Barometer Readings to o"" C, 

(/5 - a) t] 



corrected height Ho = H\l — 



(i + m 



H = observed height of barometer, 
/ = observed temperature of barometer in degrees 

Centigrade, 
j8 = 0.0001818, the coefficient of cubical expansion 

of mercury, 
a = coefl&cient of linear expansion of the material of 

the scale (0.0000085 for glass, 0.0000184 for 

brass). 

Reduction of Gaseous Volumes to o° C, and 
I Atmosphere Pressure 



corrected volume Vo 



-k 



+ 0.00367/760 



V = observed volume, 

/ = observed temperature in degrees Centigrade, 
p = pressure in millimeters of mercury. 



MEASUREMENT 167 

Determination of Altitudes by the Barometer 

For heights not exceeding 2000 feet, relative altitude 
is given by the approximate formula: 

2 (r + Ti)] H-Hi 



Z (in feet) = 52,500 1 + 



1000 H + Hi 



X = vertical distance between the two stations, 
T = Centigrade temperature at lower station, 
Ti = Centigrade temperature at upper station, 
H = height of barometer at lower station reduced to 

0° C, 
Hi = height of barometer at upper station reduced 

to 0° C. 

For any altitude, 

Z = 60,346ll+0.00256cos(2^) Jl + ^^^^Wgio^ 
in which d = latitude in degrees. 



i68 



ENGINEERING MATHEMATICS 



PHYSICAL AND CHEMICAL CONSTANTS 

ATOMIC WEIGHTS 



Element 



Aluminium . 
Antimony. . 

Argon 

Arsenic 

Barium 

Beryllium. . 
Bismuth. . . . 

Boron 

Bromine. . . . 
Cadmium. . . 
Caesium .... 
Calcium. . . . 

Carbon 

Cerium 

Chlorine. . . . 
Chromium. . 

Cobalt 

Copper 

Dysprosium. 

Erbium 

Europium. . . , 

Fluorine 

Gadolinium. 

Gallium 

Germanium. 

Gold 

Helium 

Hydrogen 

Indium 

Iodine 

Iridium 

Iron 

Krypton 

Lanthanum. . 

Lead 

Lithium 

Lutecium. . . . 
Magnesium. . 
Manganese. . 

Mercury 

Molybdenum 



Sym- 


Atomic 


bol 


weight 


Al 


27.1 


Sb 


120.2 


A 


39.88 


As 


74.96 


Ba 


137.37 


Be 


9.1 


Bi 


208.0 


B 


11.0 


Br 


79.92 


Cd 


112.40 


Cs 


132.81 


Ca 


40.07 


C 


12.00 


Ce 


140.25 


CI 


35.46 


Cr 


52.0 


Co 


58.97 


Cu 


63.57 


Dy 


162.5 


Er 


167.7 


Eu 


152.0 


F 


19.0 


Gd 


157.3 


Ga 


69.9 


Ge 


72.5 


Au 


197.2 


He 


3.99 


H 


1.008 


In 


114.8 


I 


126.92 


Ir 


193.1 


Fe 


55.84 


Kr 


82.9 


La 


139.0 


Pb 


207.10 


Li 


6.94 


Lu 


174.0 


Mg 


24.32 


Mn 


54.93 


Hg 


200.6 


Mo 


96.0 



Element 



Neodymium. . 

Neon 

Nickel . 

Niobium 

Nitrogen 

Osmium 

Oxygen 

Palladium .... 
Phosphorous. . 

Platinum 

Potassium. . . . 
Praseodymium 

Radium 

Rhodium 

Rubidium .... 
Ruthenium. . . 
Samarium. . . . 

Scandium 

Selenium 

Silicon 

Silver 

Sodium 

Strontium. . . . 

Sulphur 

Tantalum 

Tellurium 

Terbium 

Thallium 

Thorium. ..... 

Thulium 

Tin 

Titanium 

Tungsten 

Uranium 

Vanadium. . . . 

Xenon 

Ytterbium. . . . 

Yttrium 

Zinc 

Zirconium. . , 



Sym- 
bol 



Nd 

Ne 

Ni 

Nb 

N 

Os 

O 

Pd 

P 

Pt 

K 

Pr 

Ra 

Rh 

Rb 

Ru 

Sa 

Sc 

Se 

Si 

Ag 

Na 

Sr 

S 

Ta 

Te 

Tb 

Tl 

Th 

Tm 

Sn 

Ti 

W 

U 

V 

Xe 

Yb 

Y 

Zn 

Zr 



Atomic 
weight 



144.3 
20.2 
58.68 
93.5 
14.01 
190.9 

16.00 
106.7 
31.04 
195.2 
39.10 
140.6 
226.4 
102.9 
85.45 
101.7 
150.4 
44.1 
79.2 
28.3 
107.88 
23.00 
87.63 
32.07 
181.5 
127.5 
159.2 
204.0 
232.4 
168.5 
119.0 
48.1 
184.0 
238.5 
51.06 
130.2 
172.0 
89.0 
65.37 
90.6 



PHYSICAL AND CHEMICAL CONSTANTS 169 



WEIGHT AND DENSITY OF VARIOUS 
._ SUBSTANCES 

Values are for ordinary temperatures unless otherwise 

stated. 



Metals 



Aluminium 

Antimony 

Bismuth 

Brass (ordinary) 

Bronze 

Calcium 

Copper (pure) 

Copper (cast) 39° F.. 
Copper (rolled) 39° F 

Gold 

Iron (pure) 

Iron (cast) 39° F 

Iron (wrought) 39° F 

Lead 

Magnesium 

Mercury 32° F 

Nickel 

Platinum 

Potassium 

Silver 

Sodium 

Steel (hard) 39° F.. . 

Steel (soft) 39° F 

Tin 

Tungsten 

Zinc 



Weight in pounds 


per cu. in. 


per cu. ft. 


0.096 


166.5 


0.244 


422.0 


0.354 


612.0 


0.308 


532.0 


0.319 


552.0 


0.057 


98.5 


0.322 


565.0 


0.314 


541.5 


0.321 


554.0 


0.698 


1205.0 


0.284 


490.0 


0.260 


449.0 


0.281 


485.0 


0.411 


709.7 


0.064 


109.0 


0.491 


848.0 


0.318 


549.0 


0.775 


1340.0 


0.031 


53.9 


0.379 


655.0 


0.035 


60.5 


0.286 


494.0 


0.283 


488.0 


0.264 


455.0 


0.624 


1080.0 


0.253 


437.0 



Density 

relative to 

water 



2.67 
6.76 
9.82 
8.55 
8.85 
1.58 
8.93 
8.70 
8.88 

19.32 
7.86 
7.21 
7.78 

11.38 
1.75 

13.60 
8.80 

21.50 
0.87 

10.50 
0.97 
7.92 
7.83 
7.30 

17.30 
7.00 



170 



ENGINEERING MATHEMATICS 



WEIGHT AND DENSITY OF VARIOUS 
SUBSTANCES (Continued) 



Liquids 


Weight in pounds 


Density 

relative to 

water 


Acid, hydrochloric 


per cu. in. 

0.0433 

0.0440 

0.0675 

0.0286 

0.0455 

0.0455 

0.0307 

0.0332 

0.0328 

0.0307 

0.0314 

0.036121 
0.036125 
0.036085 
0.034549 
0.037023 


per cu. ft. 

74.8 
76.0 
116.5 
49.5 
78.5 
78.5 
53.0 
57.4 
56.6 
53.0 
54.1 

62.417 
62.425 
62.355 
59.700 
63.976 


1 20 


Acid, nitric 


1.22 


Acid, sulphuric 


1.84 


Alcohol 


0.79 


Carbon disulphide 


1.26 


Glycerine 


1.26 


Naphtha. . . . 


85 


Oil,, linseed 


0.92 


Oil, lubricating 


0.91 


Petroleum 


0.85 


Turpentine. ... 


87 


Water, pure, at 

32° F. (freezing point) . . . 
39.1° F. (max. density)... 
62° F. (standard temp.). . 
212° F. (boiling point)... . 
Water, Sea, 62° F 


1.0010 
1.0011 
1.0000 
0.9574 
1.0260 



Values for gases given below are for 32° F. and 

a pressure of i atmosphere. 



Gases 



Acetylene, C2H2 

Air 

Ammonia, NH3 

Carbon monoxide, CO. . 
Carbon dioxide, CO2. . . 

Ethylene, C2H4 

Hydrochloric acid, HCl. 

Hydrogen, H2 

Hydrogen sulphide, H2S 

Methane, CH4 

Nitrous oxide, N2O. . . . 

Nitric oxide, NO 

Nitrogen, N2 

Oxygen, O2 

Sulphur dioxide, SO2. . . 
Water vapor, H2O 



Weight in 


pounds 


per cu. ft. 


0.0725 


0.0807 


0.0475 


0.0781 


0.1227 


0.0781 


0.1023 


0.00562 


0.0949 


0.0446 


0.1235 


0.0831 


0.0783 


0.0892 


0.1786 


0.0502 



Density 
relative to 



0.898 

1.000 

0.589 

0.967 

1.520 

0.967 

1.268 

0.0695 

1.175 

0.553 

1.530 

1.030 

0.970 

1.105 

2.210 

0.622 



^ 



PHYSICAL AND CHEMICAL CONSTANTS 171 



WEIGHT AND DENSITY OF VARIOUS 
SUBSTANCES {Continued) 



Woods 



Ash 

Beech 

Cedar , 

Cork 

Elm 

Fir 

Lignum-litae , 
Mahogany. . , 

Maple 

Oak 

Pine, Yellow- 
Pine, White. 

'Poplar 

Spruce 

Walnut 



Weight in 
in pounds 
per cu. ft. 



45 
46 
39 
15 
38 
37 
62 
51 
42 
47 
38 
28 
30 
28 
36 



Density 

relative to 

water 



0.72 
0.73 
0.62 
0.24 
0.61 
0.59 
1.00 
0.81 
0.68 
0.75 
0.61 
0.45 
0.48 
0.45 
0.58 



Other materials 



Asphaltum 

Brick, common. . . 
Cement, average.. 

Clay 

Coal, anthracite. . 
Coal, bituminous. 
Concrete, average 

Earth, loose 

Earth, packed. . . . 
Glass, average. . . . 

Glass, flint 

Granite 

Gravel, average. . . 

Ice 

Limestone 

Marble 

Quartz 

Sand, average 

Slate 



Weight 
in pounds 
percu. ft. 



87 
112 

90 
135 

95 

84 
135 

75 
100 
164 
188 
165 
110 

56 
165 
170 
165 
100 
175 



Density 

relative to 

water 



1.39 
1.79 
1.45 
2.15 
1.50 
1.35 
2.20 
1.20 



.60 
.60 
.02 
,65 
1.75 
0.90 
2.65 
2.73 
2.65 
1.60 
2.80 



172 ENGINEERING MATHEMATICS 

MELTING AND BOILING POINTS OF ELEMENTS 



Element 


Melting point 


Boiling point at 

atmospheric 

pressure 




Degrees C. 


Degrees F. 


Degrees 



1800 
1440 
-186 
/ subl 
\450 

1420 ' 
/ subl 
\3500 
63 

778 

-33^6 
2200 

2310 ' 
-187 

2530 
-268.6 
-252.7 
184.4 

2450 

1525 

1400 

1120 

1900 
356.7 

2330 
-195.7 

-182*9 
2540 

287 
2450 

758 

690 
3500 
1955 

877 


Degrees 


Aluminium 

Antimony 

Areon 


657 
630 

-188 


1214 

1166 

-306 


3272 

2624 

-303 


Arsenic 


(volatilizes") 


imes \ 
842 1 


Barium 


850 
269 

2000 

-7.3 

321 

780 

4000 

-102 

1489 

1490 

1083 

-223 

1062 

below -2 70 

-259 

113 

1505 

327 

186 

633 

1207 

-38.8 

1452 

-210.5 

2200 

-235 

1549 

44.1 
1710 

62.5 
217 
1420 
960 
97.0 


1562 
516 

3630 

18.8 

610 

1436 

7230 

-151.5 

2712 

2714 

1982 

-370 

1944 

below— 454 

-434 

235 

2742 

621 

367 

1172 

2205 

-37.8 

2648 

-347 

3990 

-391 

2820 

111.5 
3110 
144.5 
423 
2588 
1760 
206.6 


Bismuth 


2590 


Boron 


imes \ 

6330 I 

145.5 


Bromine 


Cadmium 


1432 


Calcium 




Carbon 




Chlorine 


-28.5 


Chromium 

Cobalt 


3992 


Copper 


4190 


Fluorine 


-305 


Gold 


4586 


Helium 


-452 


Hydrogen 

Iodine 


-423 
364 


Iron 


4442 


Lead 


2776 


Lithium 


2552 


Magnesium 

Manganese 

Mercury 


2052 

3452 

674 


Nickel 


4226 


Nitrogen 


-320 


Osmium 




Oxvs^en 


-297 


Palladium 

Phosphorous 

Platinum 


4600 

549 

4440 


Potassium 

Selenium 


1397 
1274 


Silicon 


6330 


Silver 


3551 


Sodium 


1612 







^ 



PHYSICAL AND CHEMICAL CONSTANTS 1 73 



MELTING AND BOILING POINTS OF ELEMENTS 

{Continued) 



Element 


Melting point 


Boiling point at 

atmospheric 

pressure 


^ 


Degrees C. 


. Degrees F. 


Degrees 
C. 


Degrees 


Strontium 

Sulphur (rhombic) 

Tantalum 

Tin 


900 

115 
2910 

232 
2500 
3083 

418 
1300 


1650 

239 
5270 

449.6 
4530 

5582 

784 

2372 


'445' 

2270' 

3700" 
918 


'833' 
4122 


Titanium 

Tungsten 


6700 


Zinc 


1683 


Zirconium 





SPECIFIC HEATS 
The values of specific heat, unless otherwise stated, are average 
values, and hold approximately over ordinary ranges of tempera- 
tures. 



Solids 



Aluminium . . . . 

Antimony 

Bismuth 

Brass 

Copper 

Gold 

Iodine 

Iron (wrought) 
Iron (cast) . . . . 

Lead 

Magnesium . . . . 
Manganese. . . . 

Nickel 

Phosphorous. . . 

Platinum 

Silicon 

Silver 

Steel 

Sulphur 

Tin 

Tungsten 

Zinc 



Specific heat 



0.219 

0.051 

0.0304 

0.094 

0.095 

0.032 

0.054 

0.114 

0.130 

0.031 

0.246 

0.122 

0.109 

0.189 

0.033 

0.183 

0.057 

0.117 

0.203 

0.056 

0.034 

0.096 



174 



ENGINEERING MATHEMATICS 
SPECIFIC HEATS (Continued) 



Liquids 



Alcohol, methyl 

Bismuth (melted) 

Brine (density 1.2) 32° F. 

Lead (melted) 

Mercury 68° F 

Oil, olive 

Sea- water 

Sulphur (melted) 

Tin (melted) 

Turpentine 

Water 32° F 

Water 68° F 

Water 212° F 



Specific heat 



0.600 

0.0363 

0.710 

0.0402 

0.0333 

0.47 

0.94 

0.234 

0.064 

0.47 

1.0083 

0.9992 

1.0051 



Gases 



Air 

Ammonia 

Carbon monoxide 
Carbon dioxide . . 

Ethylene 

Hydrogen 

Nitrogen 

Oxygen . 



Specific heat at 
constant pressure 



0.2375 

0.508 

0.2479 

0.217 

0.404 

3.409 

0.2438 

0.2175 



Specific heat at 
constant volume 



0.1685 

0.299 

0.1758 

0.171 

0.332 

2.412 

0.1727 

0.1550 



Other materials 



Charcoal. . . . 
Glass, crown. 
Glass, flint. . 

Granite 

Ice 

India rubber. 

Marble 

Masonry 

Paraffin wax. 
Porcelain. . . . 
Quartz 



Specific heat 



0.241 

0.16 

0.12 

0.19 

0.504 

0.40 

0.21 

0.20 

0.69 

0.255 

0.18 



Note. The specific heat of a material is the number of British Thermal 
Units necessary' to raise the temperature of 1 pound of the material 1 ° F. 



PHYSICAL AND CHEMICAL CONSTANTS 1 75 

Coefficients of Linear Expansion of Solids 

The length of a soUd at any temperature is It = 
lo{l + at), lo being the known length at some given 
temperature, t the variation of temperature in degrees, 
and a the coefl&cient of linear expansion of the material. 
This formula holds approximately when the tempera- 
ture interval is not large. The coefficient of surface 
expansion equals 2 a; the coefl&cient of cubical ex- 
pansion equals 3 a. 



COEFFICIENTS OF LINEAR EXPANSION (a) 



Metals 


Forr C. 


ForTF. 


Aluminium 


0.0000222 

0.000017 

0.0000113 

0.0000176 

0.0000189 

0.0000177 

0.0000079 

0.0000160 

0.0000184 

0.0000142 

0.0000181 

0.0000100 

0.0000117 

0.0000283 

0.0000125 

0.00000863 

0.0000194 

0.0000250 

0.0000114 

0.0000209 

0.0000190 

0.0000253 


0.0000123 


Aluminium bronze 

Antimony 


0.0000095 
0.00000627 


Bismuth 


0.00000975 


Brass 


0.0000105 


Bronze 


0.00000985 


Carbon, graphite 


0.0000044 


Copper 


00000887 


German silver (120° F.). • 
Gold 


0.0000102 
00000786 


Gun metal 


0000101 


Iron (cast) 


00000556 


Iron (wrought) 


00000648 


Lead 


0000157 


Nickel 


00000695 


Platinum 


0.00000479 


Silver 


0000108 


Solder 


0.0000139 


Steel 


00000636 


Tin 

Type metal (275° F.) 

Zinc 


0.0000116 
0.0000106 
0.0000141 







176 



ENGINEERING MATHEMATICS 



COEFFICIENTS OF LINEAR EXPANSION (a) 
{Continued) 



Other materials 


For 1 ° C. 


Forl°F. 


Brick 


0.00000550 

0.0000143 

0.0000770 

0.00000850 

0.00000714 

0.00000812 

0.00000789 

0.0000507 

0.0000040 

0.0000060 

0.0000036 

0.00000050 

0.0000104 

0.0000030 
0.0000050 

0.0000600 
0.0000400 
0.0000340 


00000305 


Concrete 


00000795 


Ebonite 

Glass, soft 

Glass, hard 


0.0000428 

0.00000470 

0.00000397 


Glass, flint 


0.00000451 


Granite . . 


00000438 


Ice , 


0.0000282 


Marble 


0.0000022 


Masonry (average) 

Porcelain 


0.0000033 
0.0000020 


Silica (0°to212°F0 

Slate 


0.00000028 
0.00000577 


Woods, along grain 

beech, mahogany 

oak, pine 


0.0000017 
0.0000028 


Woods, across grain 
beech 


0.0000330 


mahogany 


0.0000220 


pine 


0.0000190 







^ 



PHYSICAL AND CHEMICAL CONSTANTS 



77 



<u o 

§ « 

o 



s 










<^ m oooofo 
ro f<^ rs — — 

CX3 0O0O0O0O 


— r^vOmfN 






of vaporiza- 
tion, 

L Y 


tPi '"f (^ OO -^ 

TfTT oo-^- 




(N<N<NOvO 

ovoor^sou-^ 
ininirMPiin 


r^^f<^mvOoo 
rr po «N — o 
iPiiominm 






o 


m sO — o^ o 


O^O^C30sO^^^ 

O^ O ' — rsi f<^ 

— CNfSfNCN 


— — vOOOt^M 
OsO — vO — 


vOOtt oorsi 


ooooo 


OOOOO 


OOOOO 


OOOOO 


;3 

1 


h 

1—1 


t>^ fO <N c<^ OO 


— 0'<r 


cAOOrr — O 


o — cN-^r^ 


c^• o^ o^ o^ o^ 


oo-^ot^-^ 


Q\ O^ O^ o^ o^ 


OOOvO-^eN 
O^ O^ O^ O^ O^ 




O^ O pO i/> c^ 


CMnrrsO — 


t^mir vOt>* 


OPor^cNoo 


■«T r<^ fN rs — 

OOOOO 


r<^ O r>. -^ (N 

OOOOO 




lilii 


pq 

1 


a 

'o 


O — -^O^vO 


NOtNu-ivO-^ 


— t^ — moo 


^ 


go 


CS <N fS fN rr, 


fTN c^, cp> (^ cr\ 


§^^55^5 






^0 


— — OmvO 


sOOOCNO-^ 


•<9-fNt>.00 


OM^^C><^ 


r^ O^ c^ rO ^■" 
tj-vCOOOnO 


§;i^jq^ 


c«^. fO ^ -"T ^ 










1 
1 


a 






os-^intsi 


r^'.r — moo 


-"f OmvO^ 


-^OfNO^OO 

— ■^ — ONO 


<N|00 — vOC<> 

vOrOfN — — 


§2S^S^ 


OOt^oO-^ — 

sovou^mm 


^^^??i 




8^§??2 


§§8^2^ 


i;^^^^^: 


^SSi?S:?R 


^5t2JQ5^ 




inoN(NvOOs 
vONOrN,r>%r^ 


<NTfr^ON«N 

OOOOOOOOOn 




1 

< 


pressure m 
inches of 
mercury 


.— <Ncc»Trir» 


vor^oooNO 


— <Nfc>Trin 


^t^^OOO^O 



178 



ENGINEERING MATHEMATICS 



s 

-t-> 

a 




hN r^ t>. t>i r>N r^r>.i>«r^ r>, t>i t>i r>. i>. t>. r>i r>. !>• r^ i>. 


of vaporiza- 
tion, 
L r 




:2- 

O 


«NfN(N<N(N fOr<^cric<^ cr> c<^ c«^ CO fO m CO cO m c^ CO 

do odd <6<6ci<6 d dddod do" odd 


1 

a 


t— 1 


'— vO — vO<N ooir><NO 00 •— oom-^co COvO — oot^ 

— c>odsdiX coVi^'d QO oomco — ov hs'roosdco' 

— oooo oooo O^ C^C^O^O^OO 0O0O0OI>>t^ 

(y^a^o^o^o^ o^a^ONOv oo oooooooooo oooooooooo 


ti 


(N — O^0Or>. vO-^COCN — •— 0^t^>r^CO — OOtOCSOs 


1 

o 
o 


i 

'o 




'B 
'o 


vooooo— ooovO"^ o ocommco o(NO'«rvO 


Volume of 
one pound 
in cu. ft., 




U 


SS2§§ SSS^J? O OCO^^<N O-00<N^ 
— — — CNCS CNtNfNtN fS «N<NfN(N<N CSJ «N <N CM CS 


2- 

< 


pressure m 
inches of 
mercury 


p. w 



PHYSICAL AND CHEMICAL CONSTANTS 179 



ctS O 

o 



03 
> 



1^ 



►^lE-. 



■3 


% 


a 


S 


■—I 


s* 











^ 




03 


Q 


u 

05 







^ 



— sO — t^ro oo-^ — r^-'T 
oo^o^oooo h>i t>. r* vo ^o 

t>. vO vO ->0 nO vO vO nO vO sO 






:S 



■^roiporru^ r>, OS — ''I- 1>, oc<^t>. — tn o^cooOf^oo 
cnfS — 00 00 t>>. r>. sO m tri -^ c<^ r<^ CN — — OOO^ 
COC<^<^C<^CN rs)(Nrv|<NrS <NfN(N<NfS CNCSCNCN — 



OS — — O^-^ t^OOoONOr^ 00<N-<rvOt^ tN.sO"^<NON 

r> c<^ 00 <N t>, — iPiavcor>. o-^t>«.opn vOo^rsimr>i 

vOr>.t>.000O O^O^OOO — — — rSfS (N<Nc<^c<^f<> 

co<^r<^cOcO cocOcO'^'^r -"I- -"I- "^ -^ "f 'tr-^'^'^'^ 

00000" 00 000 ooo'od cioooo 



m 






r>i On rs i>N rs oomt^p^— — — <N-«rtn ooomr^ — 



Ot>>itn(NO t^wn«^ — OS 

r>.sOvOsONO unmi/MnTt- 
0000000000 0000000000 



r>,unc<^'— OS r^tso-^tN — 
0000000000 0000000000 



— vOCNOstN, imrMnvOt^. 



OS — "^r^o 



0Of<M^<N 



h^-^fNOsr"* mpo — osr>. m-^cNOOs t>i u^ -"i- rsi — 
■^■^'>rfAcr» fOfomtNrq rs^r^p^^r>^— ^_^^_ 

Os OS O^ Os Os Os Os Os Os Os Os Os Os Os ^^ Os Os Os Os Os 



r>.os — csm 



c^rsrsiooo 



sO "f "-= 00 u^ •— i>.r<^osm 



sO'>ras'<rvo oooot>.in<N ooc^^r^'— m t>.os'— <s<m 

oofsmosfN LTiod-— -^r^. 

— <N<N(NfO c<^ r«^ -«r -^ '"T _ . _ _ 

fSrsJCSCNCsJ fSrslCNCNCS <NtN<NCNrs) rsl<N<NfS<N 



lag' 



som-^om — ■<)-— .r>iso f<^«NorM>.rsi oor>isot^ON 
r>.os{Nsoo movOfsoo mfNosso-^ — osf^mc^ 

rKpsjrsj — — ooo^osoo 00 00 i>i r>i r>.* t^isOsOsbsd 



O) OS 



rriO\OOs<N CNPslOOO'* O m Os <N m r^ Os Os O O 

o-^t^on- r>io<^«Xoo" — c<<imooo cs-^vdos — 

inmmvOso vo t>. t>. r>. r«i ooooooooos Osososo^o 

<Nrq<NCNCN CN rs| rsj rsj CN ps) rsj psi rsi csi cspsicsipsco 



o.ti '^ 

o d-^ . 
S5 ^ ^ o" 

< w! o 



OfN-<fsOQO 0rsl-«r\000 OC^-^sOOO OtN-^sOOO 
<r\ CO m r<^ cfS '^■•^■^Tr^ UMOmvftvrk ^>OsOnOsO 



i8o 



ENGINEERING MATHEMATICS 



2 





C^ — 0^ sO'^ 
sOsOsOsOvO 


sOsOnOvOvO 


6131 
6113 
6096 
6079 
6062 


6045 
6028 
6012 
5996 
5981 






of vaporiza- 
tion, 
L r 








Oi^r<^ot^ 


^ • " ^~ 






Is 

o 


'■*• -^ Tf -<}- TT 






vOinco — o 


ooooo 


OOOOO 


ooooo 


ooooo 



m 



rt ex 
OJ o 



H^ 



On r>. sO 'Si- CO — oo~^r>.vO 

pslfS<NrSCS <N(N — 

OOOOOOOOOO 0O0OO0< 



or^.'r — oo 
tPicorq — On 



\0 "^ ^ O^ OO 

oot>i\0"^co 



ooc«^OMn<N oomrqo^i>i -^cnooovo mcAtN — o^ 



rsi — ooot>* 

OOOOON 
0\ On On 00 OO 



vomTrrNj— ooNOOt^m 

O^ O^ O^ O^ O^ O^ OO OO OO OO 
OOOOOOOOOO OOOOOOOOOO 



w 



omomo Tj- a^ c<^ r«N »— inoN(r\r^o Tj-i>.o'^r^ 



<N(N — OOO 



vO'^'— oom 



— i^if^Noo-^ ON-^o^cor*! 



tN-»f nOOOOn >— f^iTivOOO 

i>> r*. h*. r> r*. oooooooooo 
fS<NrN|<Nrvi <NrNj<NCN<N 



O — cO-^vO 



<N rj CO CO rr> 



2§.s 



mtnoNr^iOO CNiONONtNoo 



ONOOr>.\om 



■CO<N<N — 



NONom«AiA mmirMAtn 



fl) Co 



o^oot>.irico ot>."^ — 1>. cooOfOooco oor^r^O"^ 

r^ioNor^co 

CnKNcOCOCO 

corocococo cococococo corococOfO 



^ 0) rj 

O dTJ 

w w d ;4, 

n w s CT 

!3^ ^ 2 w 

P. P, 



O<N-<rv000 OCnJ'^-nOoO OtNTTsOop SC3^^2S 

i>» t>» r>» r>N t>» OOOOOOOOOO ^onononon ooooo 



^ 



PHYSICAL AND CHEMICAL CONSTANTS l8l 





o^ 


inou^ — ho 
li-MTMrMrMn 


cooMntNoo 


intNot^m 


(Nooor^in 
vomrofN — 
r>i !>«. t^ t^ i>i 






of vaporiza- 
tion, 
L r 


oo sO ■<*• (^ rsj 


2? iO s::? 2^ )fi 

sss§§ 




ooooo 








o 






on-Or^oO 




ooooo 


ooooo 


OOOOO 


OOOOO 



'S Q. 


vO-^t^r^JO 


0^00 00^>^0 


voinmm'.j- 


TT-^mmin 




iiiSg 


O^ O^ O^ O^ Q^ 

t>. t»» r>. t>« r>. 


t>i hs r>. t>i t>i 


ocoooooooo 


7 ^< 


oot>.r^\ONO 


inmtn-^-^ 




vOvOt^OOOO 


rt J-i 


OOoOOoSoO 
OOOOOOOOOO 


g|g|g 


rrrPitN — O 





m 



OJ^sOOO — 



TfvOO — -"T 



\oo^ — cAvn r>.o^ — fOtn 



— u^CXNvO 



C?^ tN -ra- t>. ON <Ntj-vOOOO 



«Nf<MnvOI>. 



fOmc<^roro r<^ co r<^ r<^ r<^ 



■<!MnvOt>iOO 



^1 

B 

:3 









r«*.oo — h^m mvoovom >— <ncoivoo vor^oo^o* 

inoofNu^cs c<^t>«.(NvO— sO — vo — r>. rsjoO'«roMn 

oo^o^oot>. r^vONOmm ^j- tj- co co <n rg — — oo 

■^mr«^r<^r<^ c^mmcAm c<^f<^f<^r<^m r<\ cr<i trs tr<, cr^ 






OO — -^hoO 



■\t^o<N ■>!»• m r>« oo o ^-cNtTNfA-^ 

■^sOh*.QOo — {Nfomsd h>ooo^o(N ro"«rmsor^ 

po CO c<^ r<^ ■"^ ■^■^■^-^rf TT^-^unm vnu-Momm 

cocococOfO r<^ CO f<^ ro c<^ cor<^comco c^ co c^ c<> f<^ 



o stj . 

P. P* 



— — — — — rqrsieN(Nrg 



l82 



ENGINEERING MATHEMATICS 



p. 



o «o 

rt o 



O 



c 



t>, vO nO >0 ^ nO sO vO vO 'O tn l/M/MrMJ^i lA ITMO ITMH 

loinmtnm mirnrM/Mn i/Mommm mmirMrMr> 



r>i"^fNOtN, intMOoONO f<^ — o^r>»«r> rri — oovO-n- 
inmmu-*. -^ rf-^j-'ifcoro cot^tNtNCsj cscs — — — 
ooooo ooooo ooooo ooooo 



— — — — — — r^^r^^r^^r^^ 

lA min m u^ irMTM/Mn lA 



ooo — f<Mn voooo^o — 

int^iooo^o — rspTMAsO 

<^r^JtN<Nm fAC<>fAf<>fA 

iTMniAiAio mmmmiA 



ooooo ooooo 



ooooo 



ooooo 



pq 



2 Q. 



1^ 



vOvOt^t>iOO ONO — (NfA -^J-iAvOOOO^ OtN-^mt^ 

— c><^ooi>^ vOvOin-^fA fs^o 0^*00 odr^.sOin-^ 

oo oo t>. t>. h". !>«. r>. r>i t^ h*. t>i t>. r>. \0 vo nOvOsOnOvO 

t>. i>« r>. r« r>i i>. t>. r>. t>« r>» i>» t>. t>» i>. r>. t>, t>i r>. r^ r>i 



o^o — (^-^ mvor^o^o (nc^mavooo 



vO ^^5 vQ \0 ^-O 
OOOOOOOOOO 



vOmtriiA«A in, lAtOiAiA 
OOOOOOOOOO OOOOOOOOOO 



OCN-'MOt^ 

ro — oo^od 

OOOOi 






&>. 



t>iO^»— pom 



t>.OOOfSPA 



lAvOOOO^^- <N-«*-iAnOOO 



OOO^O — — (Nr^fAfTNfrs rri f^i (Ts rr> r<^ «NP^1 — — O 

,00O>> O— (NfArf' lAvdl^OOOv 

icAfA ■^'f-^'^'^ -^ •«!»■ "^ -^ ^r 

I (^ c<\ CO fA cA CO fO fO CO c^ f<^ rn 






O O^ O^ O^ OO 
CO <N fsi <N CN 



OO0Ot>*l^t^ 
«N <N <N fS <N 



vO v6 ^ tA«/S 
CS <N <N r^ CN 



mm' 

<N Psi <N <N <N 



<U TO 



mmiAvONO vOvOsomm u^-^-^cocm 



•OO^OOI^ 



ooa^o — r^ 

lA lA ■O vO nO 
CA CA CA CA CA 



CA CA CA CA CA 



OOO^O — <N PO-^-^iAvO 

sOvor>.r^i>« r>. tv. r>. r>i t>i 

CACACACACA CACACACACA 



d ^ 

o :^'d . 
ti <^ X ^ 



QtN'n'vOOO QfN-n-sOOO OtN-^vOOO Q <N 3- vO OO 
lA lA lA lA tA ^o nO vO vO vO t>. t>> l>>. I>« t^ OOOOOOOOOO 



PHYSICAL AND CHEMICAL CONSTANTS 183 



o 
u 





<o 
0^ 


ilisl 


u-mt, vn ir> m 


r<^ <^ <^ fN (N 








of vaporiza- 
tion, 
L r 


00000 




<NOt^vOin 

OvO — t^C<^ 






—0000 


00000 


00000 




ft 




f^So^o — 


•^ rj- ■r^ iTMn 
ir\ yr\ yr\ lA i/> 


mm tn vc sO 


<rNmsOt^^<. 

sOOOOfN-^ 

m m m m i/> 


00000 


00000 


00000 


00000 








1 ^ 

a; 


OsO<N'<l-vO 


OOOOO^O — 


fNrTMni^ON 


(Nmt^o-i- 


W 


sOvOsOvOsO 


m mmmm 


^^^5^ 


^^P^^;?; 













Ov — <^vO00 O — <Nc<Mn 



NOOOOfOiTi 



00 — "Tr^.- 



00 00 00 oo 00 



-^fNOOOsO 

"«r ■<*■ -"a- c<^ c<^ 
0000000000 



0000000000 0000000000 



o^o — p^^"^ inr^o<svr» t<sO^'— f*Mr» noooo — <n 

t>» 00 00 00 OO 00 00 CT^ O^ O^ O^ O^ O O O O O ^ *" ^^ 

CT^o^o^o^o^ CTvO^'O^o^^ CT^cooo 00000 

— — — — — _____ __(>sjrN)(N «N(N<N<^fS 



oovoor^vo iAt>.oooo — — — 00 o^r^vo-^c^ 

OO'— <NfO •^sood — r^' mr>.'o^--fO "^^oos^ 

iTMO m m lO m «r> in vo >o v© vO vO r>. i*». t^t>»r^oOQO 

CO c<^ c<^ c<^ r<^ cAfOr<^fOc<^ ror<^cororo rororoc<^f<^ 



I §.2 



m<N — — po 

"^OsOCSOO 
OOO^O^OC 



vo — r^m po 



<N«NPsJCN<N CNtNtNCsJCN 






vomcAfNO o^ovooo o^oovomf<>i — OMorsio^ 

r>.ooo^O— — f<^vdooo — r<MrM>.o — <N-^vOf>. 

t>» t>« (>• 00 oo 00 00 oO oO O"^ o^ o^ o^ o^ o^ o o o o o 

c<^ rO CO CO CO CO CO cO CO cO co co co co co ^ ■^ ^ ■^ rf 



<u.S S3 
o md . 

p. p« 



OfS-'J-NOOO Otr»OiA< 
O^ C^ O^ CT^ On O O "— ^ < 



Ovoomo 

CNcOCO-^-er «rMONO>Ot>* 



i84 



ENGINEERING MATHEMATICS 



2 






K|E-< 



OS 



o 



r>.mcofNo o^sOro 
—.— — — — ooo 
in i/^ lA ir> un iTMn u^ 






O^ ^^ O^ O^ O^ O^ ^^ O*^ O^ OO 



oooooooooo oooooor>it>. 



ooooo ooooo ooooo ooooo 



ooooo ooooo ooooo ooooo 



S o 






(N — o^ r>i sO '«»■— od in rsi o^ sO -^ — od m m o od t>. 

r<^m<N<N«N rvirsl — — — OOOOO O^OOt>."^fS 

r^ r^ 1**^ r^ t^ r^ r^ t^ r*% r*% 1*% t^ r^ t^ ^<o ^^o ^^3 ^^o ^'O ^^5 

■"ToocNvoo ■^■^(^m-^ msooo — en so-^ooom 

r^inrr^N— o^vdcnor*. ■^'— odvom or^m — o 

— .— — — — ooooo o^o^oooooo oc vO m cn — 

oooocx)ooco oooooooor*. t^ r^, r>. t>. t>. h*. t>, t>. i>. r>. 



t3 



■«f in vO tN, CO ovofscn-^ inm^ovOvO mrm^oo'^ 



— — — — — — fses<NtN 

ooooo ooooo 

fNfSCNfNCS tN fS fS fN r^ 



OOOOO oooo^o^ 



<:>ts,Tj- — OO -'i-h^ia^oo 



mm r>iO>> o 

00 0O0OC30ON 

rn cn cn cn cn 



c^i in OO fs in 



ooM>.infn 
OO o cn vO o^ 



OOOvOO — 



<NTj-vOOOt>. 



•Si 

O 3 



a 



^ a; ^ 



in-^-'T -^cn 



cn<Nr>4«N— — OO^t>.vO 






sOr^ioo-^o 



mm ^cn 



ONvO<Noocn 



oomtNm-^ 

■«t- vO t>.' vd cn 



o^ — (N'^in r>.ocnvooN — ■'j-rs.ovcN ■«»■ vO t>. vO cn 
o — — — — ■ — cN(N(Nc^i cncncncn-^ -^m-oooo 
■^■^'tr-n-'^ ■^■"T'^'^'^ ■^■^'^■^•■^ TT-^TTTrm 






w 



d 



momom oooOQ ooooo QOQCQ 
r<»ooooo^a^ o — f^cn^ m^or^iooo^ QmoQO 
(NfNrs<NCN c^mcncncn cncncncncn ■^■«rmvor>» 



^^ 



L 




'850 



900 950 100O 

HEAT CONTENT, B.T.U. 



1050 



MOLLIER S 
Reproduced with permission from G. A. Goodenoug' 

Messrs. Joh( 



13S0 



1450. 



2.10 




1200 1250 1300 1390 1400 1450 



1.30 



AM CHART 

"Properties of Steam and Ammonia," published by 
iley & Sons. 



i 



w 



TABLES 



I8S 



TABLES 



CIRCUMFERENCES AND AREAS OF CIRCLES 



' Diam- 


Circum- 


Area 


Diam- 


Circum- 


Area 


eter 


ference 


eter 


ference 


1 


3.1416 


0.7854 


26 


81.681 


530.93 


2 


6 2832 


3.1416 


27 


84.823 


572.56 


3 


9.4248 


7.0686 


28 


87.965 


615.75 


4 


12.5664 


12.5664 


29 


91.106 


660.52 


5 


15.7080 


19.635 


30 


94.248 


706.86 


6 


18.850 


28.274 


31 


97.389 


754.77 


7 


21.991 


38.485 


32 


100.53 


804.25 


8 


25.133 


50.266 


33 


103.67 


855.30 


9 


28.274 


63.617 


34 


106.81 


907.92 


10 


31.416 


78.540 


35 


109.96 


962.11 


11 


34.558 


95.033 


36 


113.10 


1017.88 


12 


37.699 


113.10 


37 


116.24 


1075.21 


13 


40.841 


132.73 


38 


119.38 


1134.11 


14 


43.982 


153.94 


39 


122.52 


1194.59 


15 


47.124 


176.71 


40 


125.66 


1256.64 


16 


50.265 


201.06 


41 


128.81 


1320.25 


17 


53.407 


226.98 


42 


131.95 


1385.44 


18 


56.549 


254.47 


43 


135.09 


1452.20 


19 


59.690 


283.53 


44 


138.23 


1520.53 


20 


62.832 


314.16 


45 


141.37 


1590.43 


21 


65.973 


346.36 


46 


144.51 


1661 .90 


22 


69.115 


380.13 


47 


147.65 


1734.94 


23 


72.257 


415.48 


48 


150.80 


1809.56 


24 


75.398 


452.39 


49 


153.94 


1885.74 


25 


78.540 


490.87 


50 


157.08 


1963.50 



Note. — The surface of a sphere of given diameter may be found directly 
from the above table, since it is equal to the area of a circle of twice the diam- 
eter of the sphere. 



i86 



ENGINEERING MATHEMATICS 



CIRCUMFERENCES AND AREAS OF CIRCLES 

( Continued) 



Diam- 
eter 

51 

52 
53 
54 
55 
56 
5.7 
58 
59 
60 
61 
62 
63 
64 
65 
66 
67 
68 
69 
70 
71 
72 
73 
74 
75 



Circum- 


ference 


160.22 


163.36 


166.50 


169.65 


172.79 


175.93 


179.07 


182.21 


185.35 


188.50 


191.64 


194.78 


197.92 


201.06 


204.20 


207.34 


210.49 


213.63 


216.77 


219.91 


223.05 


226.19 


229.34 


232.48 


235.62 



Area 



2042.82 
2123.72 
2206.18 
2290.22 
2375.83 
2463.01 
2551.76 
2642.08 
2733.97 
2827.43 
2922.47 
'3019.07 
3117.25 
3216.99 
3318.31 
3421.19 
3525.65 
3631.68 
3739.28 
3848.45 
3959 . 19 
4071.50 
4185.39 
4300.84 
4417.86 



Diam- 


Circum- 


eter 


ference 


76 


238.76 


77 


241.90 


78 


245.04 


79 


248 . 19 


80 


251.33 


81 


254.47 


82 


257.61 


83 


260.75 


84 


263.89 


85 


267 .04 


86 


270.18 


87 


273.32 


88 


276.46 


89 


279.60 


90 


282.74 


91 


285.88 


92 


289.03 


93 


292.17 


94 


295.31 


95 


298.45 


96 


301.59 


97 


304.73 


98 


307.88 


99 


311.02 


100 


314.16 



Area 



4536.46 
4656.63 
4778.36 
4901.67 
5026.55 
5153.00 
5281 .02 
5410.61 
5541.77 
5674.50 
5808.80 
5944.68 
6082.12 
6221.14 
6361.73 
6503.88 
6647.61 
6792.91 
6939.78 
7088.22 
7238.23 
7389.81 
7542.96 
7697.69 
7853.98 



TABLES 187 

POWERS, ROOTS, AND RECIPROCALS 



Number 


Square 


Cube 


Square root 


Cube root 


Reciprocal 


1 


1 


1 


1.000000 


1.000000 


1.0000000 


2 


4 


8 


1.414214 


1.259921 


.5000000 


3 


9 


27 


1.732051 


1.442250 


.3333333 


4 


16 


64 


2.000000 


1.587401 


.2500000 


5 


25 


125 


2.236068 


1.709976 


.2000000 


6 


36 


216 


2.449490 


1.817121 


.1666667 


7 


49 


343 


2.645751 


1.912931 


.1428571 


8 


64 


512 


2.828427 


2.000000 


. 1250000 


9 


81 


729 


3.000000 


2.080084 


.1111111 


10 


100 


1000 


3.162278 


2 . 154435 


.1000000 


11 


121 


1331 


3.316625 


2.223980 


.0909091 


12 


144 


1728 


3.464102 


2.289429 


m33333 


13 


169 


2197 


3.605551 


2.351335 


.0769231 


14 


196 


2744 


3.741657 


2.410142 


.0714286 


15 


225 


3375 


3.872983 


2.466212 


.0666667 


16 


256 


4096 


4.000000 


2.519842 


.0625000 


17 


289 


4913 


4.123106 


2.571282 


.0588235 


18 


324 


5832 


4.242641 


2.620741 


.0555556 


19 


361 


6859 


4.358899 


2.668402 


.0526316 


20 


400 


8000 


4.472136 


2.714418 


.0500000 


21 


441 


9261 


4.582576 


2.758924 


.0476190 


22 


484 


10,648 


4.690416 


2.802039 


.0454545 


23 


529 


12,167 


4.795832 


2.843867 


.0434783 


24 


576 


13,824 


4.898980 


2.884499 


.0416667 


25 


625 


15,625 


5.000000 


2.924018 


.0400000 


26 


676 


17,576 


5.099020 


2.962496 


.0384615, 


27 


729 


19,683 


5.196152 


3.000000 


.0370370 


28 


784 


21,952 


5.291503 


3.036589 


.0357143 


29 


841 


24,389 


5.385165 


3.072317 


.0344828 


30 


900 


27,000 


5.477226 


3.107233 


.0333333 


31 


961 


29,791 


5.567764 


3.141381 


.0322581 


32 


1024 


32,768 


5.656854 


3.174802 


.0312500 


33 


1089 


35,937 


5.744563 


3.207534 


.0303030 


34 


1156 


39,304 


5.830952 


3.239612 


.0294118 


35 


1225 


42,875 


5.916080 


3.271066 


.0285714 


36 


1296 


46,656 


6.000000 


3.301927 


.0277778 


37 


1369 


50,653 


6.082763 


3.332222 


.0270270 



i88 



ENGINEERING MATHEMATICS 



POWERS, ROOTS, AND RECIPROCALS 

{Continued) 



Number 


Square 


Cube 


Square root 


Cube root 


Reciprocal 


38 


1444 


54,872 


6.164414 


3.361975 


.0263158 


39 


1521 


59,319 


6.244998 


3.391211 


.0256410 


40 


1600 


64,000 


6.324555 


3.419952 


.0250000 


41 


1681 


68,921 


6.403124 


3.448217 


.0243902 


42 


1764 


74,088 


6.480741 


3.476027 


.0238095 


43 


1849 


79,507 


6.557439 


3.503398 


.0232558 


44 


1936 


85,184 


6.633250 


3.530348 


.0227273 


45 


2025 


91,125 


6.708204 


3.556893 


.0222222 


46 


2116 


97,336 


6.782330 


3.583048 


.0217391 


47 


2209 


103,823 


6.855655 


3.608826 


.0212766 


48 


2304 


110,592 


6.928203 


3.634241 


.0208333 


49 


2401 


117,649 


7.000000 


3.659306 


.0204082 


50 


2500 


125,000 


7.071068 


3.684031 


.0200000 


51 


2601 


132,651 


7.141428 


3.708430 


.0196078 


52 


2704 


140,608 


7.211103 


3.732511 


.0192308 


53 


2809 


148,877 


7.280110 


3.756286 


.0188679 


54 


2916 


157,464 


7.348469 


3.779763 


.0185185 


55 


3025 


166,375 


7.416199 


3.802953 


.0181818 


56 


3136 


175,616 


7.483315 


3.825862 


.0178571 


57 


3249 


185,193 


7.549834 


3.848501 


.0175439 


58 


3364 


195,112 


7.615773 


3.870877 


.0172414 


59 


3481 


205,379 


7.681146 


3.892997 


.0169492 


60 


3600 


216,000 


7.745967 


3.914868 


.0166667 


61 


3721 


226,981 


7.810250 


3.936497 


.0163934 


. 62 


3844 


238,328 


7.874008 


3.957892 


.0161290 


63 


3969 


250,047 


7.937254 


3.979057 


.0158730 


64 


4096 


262,144 


8.000000 


4.000000 


.0156250 


65 


4225 


274,625 


8.062258 


4.020726 


.0153846 


66 


4356 


287,496 


8.124038 


4.041240 


.0151515 


67 


4489 


300,763 


8.185353 


4.061548 


.0149254 


68 


4624 


314,432 


8.246211 


4.081655 


.0147059 


69 


4761 


328,509 


8.306624 


4.101566 


.0144928 


70 


4900 


343,000 


8.366600 


4.121285 


.0142857 


71 


5041 


357,911 


8.426150 


4.140818 


.0140845 


72 


5184 


373,248 


8.485281 


4.160168 


.0138889 


73 


5329 


389,017 


8.544004 


4.179339 


.0136986 



TABLES 189 

POWERS, ROOTS, AND RECIPROCALS (Continued) 



Number 


Square 


Cube 


Square root 


Cube root 


Reciprocal 


74 


5476 


405.224 


8.602325 


4 . 198336 


.0135135 


75 


5625 


421,875 


8.660254 


4.217163 


.0133333 


76 


5776 


438,976 


8.717798 


4.235824 


.0131579 


77 


5929 


456,533 


8.774964 


4.254321 


.0129870 


78 


6084 


474,552 


8.831761 


4.272659 


.0128205 


79 


6241 


493,039 


8.888194 


4.290840 


.0126582 


80 


6400 


512,000 


8.944272 


4.308870 


.0125000 


81 


6561 


531,441 


9.000000 


4.326749 


.0123457 


82 


6724 


551,368 


9.055385 


4.344482 


.0121951 


83 


6889 


571,787 


9.110434 


4.362071 


.0120482 


84 


7056 


592,704 


9.165151 


4.379519 


.0119048 


85 


7225 


614,125 


9.219545 


4.396830 


.0117647 


86 


7396 


636,056 


9.273619 


4.414005 


.0116279 


87 


7569 


658,503 


9.327379 


4.431048 


.0114943 


88 


7744 


681,472 


9.380832 


4.447960 


.0113636 


89 


7921 


704,969 


9.433981 


4.464745 


.0112360 


90 


8100 


729,000 


9.486833 


4.481405 


.0111111 


91 


8281 


753,571 


9.539392 


4.497941 


.0109890 


92 


8464 


778,688 


9.591663 


4.514357 


0108696 


93 


8649 


804,357 


9.643651 


4.530655 


.0107527 


94 


8836 


830,584 


9.695360 


4.546836 


.0106383 


95 


9025 


857,375 


9.746794 


4.562903 


.0105263 


96 . 


9216 


884,736 


9.797959 


4.578857 


.0104167 


97 


9409 . 


912,673 


9.848858 


4.594701 


.0103093 


98 


9604 


941,192 


9.899495 


4.610436 


.0102041 


99 


9801 


970,299 


9.949874 


4.626065 


.0101010 


100 


10,000 


1,000,000 


10.000000 


4.641589 


.0100000 



Logarithmic Cross- section Paper 

Cross-section paper the rulings of which are pro- 
portional to the logarithms of the scale is called loga- 
rithmic cross-section paper. This paper is most con- 
venient for plotting equations with constant exponents 
since they are straight lines on logarithmic paper while 



190 ENGINEERING MATHEMATICS 

'they are curves if plotted on ordinary graph paper, in 
which case they must be plotted point by point. 

The chief use of logarithmic cross-section paper is for 
plotting equations of the form: 

y = ax^ 

If two pairs of values of x and y are known, the corre- 
sponding points may be plotted on logarithmic paper 
and joined by a straight line. The value of the coeffi- 
cient a is equal to the intercept of this Une on the F-axis, 
and the value of the exponent n is equal to the slope of 
the line (that is, the tangent of the angle which the line 
makes with the X-axis). The reason for this is that 
plotting on logarithmic paper is equivalent to taking 
logarithms, in which case we would obtain: 

log y = loga + n log x 

which is the equation of a straight line, log a being the 
intercept and n the slope. 

In case the values of a and n are known, that is, the 
intercept and the slope, we may plot the line, and from 
it obtain any pair of values of x and y. 

Use of Logarithm Tables 

Every logarithm consists of two parts: a positive or 
negative whole number called the characteristic, and a 
positive fraction, called the mantissa. The mantissa 
is always expressed as a decimal, and is the part which 
is given in the tables. 
To find the common logarithm of a given number: 
If the number is greater than 1, the characteristic 
of the logarithm is one unit less than the number of fig- 
ures on the left of the decimal point. 



TABLES 191 

If the number is less than 1, the characteristic of the 
logarithm is negative, and one unit more than the 
number of zeros between the decimal point and the first 
significant figure of the given nimiber. 
Thus, 

log 20.6 = 1.3139 (base 10) 

log 2.06 = 0.3139 

log 0.206 = 0.3139 - 1 = 9.3139 - 10 
log 0.0206 = 0.3139 - 2 = 8.3139 - 10 

To find the number corresponding to a given 
common logarithm : 

If the characteristic of a given logarithm is posi- 
tive, the number of figures in the integral part of the 
corresponding number is one more than the number of 
units in the characteristic. 

If the characteristic is negative, the number of zeros 
between the decimal point and the first significant 
figure of the corresponding number is one less than the 
number of units in the characteristic. 



192 



ENGINEERING MATHEMATICS 



COMMON LOGARITHMS OF NUMBERS 
(Base 10) 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


0000 


0043 


0086 


0128 


0170 


0212 


0253 


0294 


0334 


0374 


11 


0414 


0453 


0492 


0531 


0569 


0607 


0645 


0682 


0719 


0755 


12 


0792 


0828 


0864 


0899 


0934 


0969 


1004 


1038 


1072 


1106 


13 


1139 


1173 


1206 


1239 


1271 


1303 


1335 


1367 


1399 


1430 


14 


1461 


1492 


1523 


1553 


1584 


1614 


1644 


1673 


1703 


1732 


15 


1761 


1790 


1818 


1847 


1875 


1903 


1931 


1959 


1987 


2014 


16 


2041 


2068 


2095 


2122 


2148 


2175 


2201 


2227 


2253 


2279 


17 


2304 


2330 


2355 


2380 


2405 


2430 


2455 


2480 


2504 


2529 


18 


2553 


2577 


2601 


2625 


2648 


2672 


2695 


2718 


2742 


2765 


19 . 


2788 


2810 


2833 


2856 


2878 


2900 


2923 


2945 


2967 


2989 


20 


3010 


3032 


3054 


3075 


3096 


3118 


3139 


3160 


3181 


3201 


21 


3222 


3243 


3263 


3284 


3304 


3324 


3345 


3365 


3385 


3404 


22 


3424 


3444 


3464 


3483 


3502 


3522 


3541 


3560 


3579 


3598 


23 


3617 


3636 


3655 


3674 


3692 


3711 


3729 


3747 


3766 


3784 


24 


3802 


3820 


3838 


3856 


3874 


3892 


3909 


3927 


3945 


3962 


25 


3979 


3997 


4014 


4031 


4048 


4065 


4082 


4099 


4116 


4133 


26 


4150 


4166 


4183 


4200 


4216 


4232 


4249 


4265 


4281 


4298 


27 


4314 


4330 


4346 


4362 


4378 


4393 


4409 


4425 


4440 


4456 


28 


4472 


4487 


4502 


4518 


4533 


4548 


4564 


4579 


4594 


4609 


29 


4624 


4639 


4654 


4669 


4683 


4698 


4713 


4728 


4742 


4757 


30 


4771 


4786 


4800 


4814 


4829 


4843 


4857 


4871 


4886 


4900 


31 


4914 


4928 


4942 


4955 


4969 


4983 


4997 


5011 


5024 


5038 


32 


5051 


5065 


5079 


5092 


5105 


5119 


5132 


5145 


5159 


5172 


33 


5185 


5198 


5211 


5224 


5237 


5250 


5263 


5276 


5289 


5302 


34 


5315 


5328 


5340 


5353 


5366 


5378 


5391 


5403 


5416 


5428 


35 


5441 


5453 


5465 


5478 


5490 


5502 


5514 


5527 


5539 


5551 


36 


5563 


5575 


5587 


5599 


5611 


5623 


5635 


5647 


5658 


5670 


37 


5682 


5694 


5705 


5717 


5729 


5740 


5752 


5763 


5775 


5786 


38 


5798 


5809 


5821 


5832 


5843 


5855 


5866 


5877 


5888 


5899 


39 


5911 


5922 


5933 


5944 


5955 


5966 


5977 


5988 


5999 


6010 


40 


6021 


6031 


6042 


6053 


6064 


6075 


6085 


6096 


6107 


6117 


41 


6128 


6138 


6149 


6160 


6170 


6180 


6191 


6201 


6212 


6??? 


42 


6232 


6243 


6253 


6263 


6274 


6284 


6294 


6304 


6314 


6325 


43 


6335 


6345 


6355 


6365 


6375 


6385 


6395 


6405 


6415 


6425 


44 


6435 


6444 


6454 


6464 


6474 


6484 


6493 


6503 


6513 


6522 


45 


6532 


6542 


6551 


6561 


6571 


6580 


6590 


6599 


6609 


6618 


46 


6628 


6637 


6646 


6656 


6665 


6675 


6684 


6693 


6702 


6712 


47 


6721 


6730 


6739 


6749 


6758 


6767 


6776 


6785 


6794 


6803 


48 


6812 


6821 


6830 


6839 


6848 


6857 


6866 


6875 


6884 


6893 


49 


6902 


6911 


6920 


6928 


6937 


6946 


6955 


6964 


6972 


6981 


50 


6990 


6998 


7007 


7016 


7024 


7033 


7042 


7050 


7059 


7067 


51 


7076 


7084 


7093 


7101 


7110 


7118 


7126 


7135 


7143 


7152 


52 


7160 


7168 


7177 


7185 


7193 


7202 


7210 


7218 


7226 


7235 


53 


7243 


7251 


7259 


7267 


7275 


7284 


7292 


7300 


7308 


7316 


54 


7324 


7332 


7340 


7348 


7356 


7364 


7372 


7380 


7388 


7396 



TABLES 



193 



COMMON LOGARITHMS OF NUMBERS 
(Continued) 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


55 


7404 


7412 


7419 


7427 


7435 


7443 


7451 


7459 


7466 


7474 


56 


7482 


7490 


7497 


7505 


7513 


7520 


7528 


7536 


7543 


7551 


57 


7559 


7566 


7574 


7582 


7589 


7597 


7604 


7612 


7619 


7627 


58 


7634 


7642 


7649 


7657 


7664 


7672 


7679 


7686 


7694 


7701 


59 


7709 


7716 


7723 


7731 


7738 


7745 


7752 


7760 


7767 


7774 


60 


7782 


7789 


7796 


7803 


7810 


7818 


7825 


7832 


7839 


7846 


61 


7853 


7860 


7868 


7875 


7882 


7889 


7896 


7903 


7910 


7917 


62 


7924 


7931 


7938 


7945 


7952 


7959 


7966 


7973 


7980 


7987 


63 


7993 


8000 


8007 


8014 


8021 


8028 


8035 


8041 


8048 


8055 


64 


8062 


8069 


8075 


8082 


8089 


8096 


8102 


8109 


8116 


8122 


65 


8129 


8136 


8142 


8149 


8156 


8162 


8169 


8176 


8182 


8189 


66 


8195 


8202 


8209 


8215 


8222 


8228 


8235 


8241 


8248 


8254 


67 


8261 


8267 


8274 


8280 


8287 


8293 


8299 


8306 


8312 


8319 


68 


8325 


8331 


8338 


8344 


8351 


8357 


8363 


8370 


8376 


8382 


69 


8388 


8395 


8401 


8407 


8414 


8420 


8426 


8432 


8439 


8445 


70 


8451 


8457 


8463 


8470 


8476 


8482 


8488 


8494 


8500 


8506 


71 


8513 


8519 


8525 


8531 


8537 


8543 


8549 


8555 


8561 


8567 


72 


8573 


8579 


8585 


8591 


8597 


8603 


8609 


8615 


8621 


8627 


73 


8633 


8639 


8645 


8651 


8657 


8663 


8669 


8675 


8681 


8686 


74 


8692 


8698 


8704 


8710 


8716 


8722 


8727 


8733 


8739 


8745 


75 


8751 


8756 


8762 


8768 


8774 


8779 


8785 


8791 


8797 


8802 


76 


8808 


8814 


8820 


8825 


8831 


8837 


8842 


8848 


8854 


8859 


77 


8865 


8871 


8876 


8882 


8887 


8893 


8899 


8904 


8910 


8915 


78 


8921 


8927 


8932 


8938 


8943 


8949 


8954 


8960 


8965 


8971 


79 


8976 


8982 


8987 


8993 


8998 


9004 


9009 


9015 


9020 


9025 


80 


9031 


9036 


9042 


9047 


9053 


9058 


9063 


9069 


9074 


9079 


81 


9085 


9090 


9096 


9101 


9106 


9112 


9117 


9122 


9128 


9133 


82 


9138 


9143 


9149 


9154 


9159 


9165 


9170 


9175 


9180 


9186 


83 


9191 


9196 


9201 


9206 


9212 


9217 


9222 


9227 


9232 


9238 


84 


9243 


9248 


9253 


9258 


9263 


9269 


9274 


9279 


9284 


9289 


85 


9294 


9299 


9304 


9309 


9315 


9320 


9325 


9330 


9335 


9340 


86 


9345 


9350 


9355 


9360 


9365 


9370 


9375 


9380 


9385 


9390 


87 


9395 


9400 


9405 


9410 


9415 


9420 


9425 


9430 


9435 


9440 


88 


9445 


9450 


9455 


9460 


9465 


9469 


9474 


9479 


9484 


9489 


89 


9494 


9499 


9504 


9509 


9513 


9518 


9523 


9528 


9533 


9538 


90 


9542 


9547 


9552 


9557 


9562 


9566 


9571 


9576 


9581 


9586 


91 


9599 


9595 


9600 


9605 


9609 


9614 


9619 


9624 


9628 


9633 


92 


9638 


9643 


9647 


9652 


9657 


9661 


9666 


9671 


9675 


9680 


93 


9685 


9689 


9694 


9699 


9703 


9708 


9713 


9717 


9722 


9727 


94 


9731 


9736 


9741 


9745 


9750 


9754 


9759 


9763 


9768 


9773 


95 


9777 


9782 


9786 


9791 


9795 


9800 


9805 


9809 


9814 


9818 


96 


9823 


9827 


9832 


9836 


9841 


9845 


9850 


9854 


9859 


9863 


97 


9868 


9872 


9877 


9881 


9886 


9890 


9894 


9899 


9903 


9908 


98 


9912 


9917 


9921 


9926 


9930 


9934 


9939 


9943 


9948 


9952 


99 


9956 


9961 


9965 


9969 


9974 


9978 


9983 


9987 


9991 


9996 



194 



ENGINEERING MATHEMATICS 



NATURAL LOGARITHMS OF NUMBERS FROM 
1 TO 10 (Base e) 



0.0000 
0.0953 
0.1823 
0.2624 
0.3365 

0.4055 
0.4700 
0.5306 
0.5878 
0.6419 



0.0099 
0.1044 
0.1906 
0.2700 
0.3436 

0.4121 
0.4762 
0.5365 
0.5933 
0.6471 



0.6932 0.6981 
0.7419 0.7467 



0.7885 
0.8329 
0.8755 

0.9163 
0.9555 
0.9933 
1.0296 
1.0647 

1.0986 
1.1314 
1.1632 
1.1939 
1.2238 

1.2528 
1.2809 
1.3083 
1.3350 
1.3610 

1.3863 
1.4110 
1.4351 
1.4586 
1.4816 

1.5041 
1.5261 
1.5476 
1.5686 
1.5892 



1.6094 
1.6292 
1.6487 
1.6677 
1.6864 



0.7950 
0.8373 
0.8796 

0.9203 
0.9594 
0.9970 
1.0332 
1.0681 

1.1019 
1.1346 
1.1663 
1.1970 
1.2267 

1.2556 
1.2837 
1.3110 
1.3376 
1.3635 



1.3888 
1.4134 
1 4375 
1.4609 
1.4839 



1.5063 
1.5282 
1.5497 
1.5707 
1.5913 



1.6114 
1.6312 
1.6506 



0.0198 
0.1133 
0.1989 
0.2776 
0.3507 

0.4187 
0.4824 
0.5423 
0.5988 
0.6523 

0.7031 
0.7514 
0.7975 
0.8416 
0.8838 

0.9243 
0.9632 
1.0006 
1.0367 
1.0716 

1.1053 
1.1378 
1 1694 
1.2000 
1.2296 

1.2585 
1.2865 
1.3137 
1.3403 
1.3661 



1.3913 
1.4159 
1.4398 
1.4633 
1.4861 

1.5085 
1.5304 
1.5518 
1.5728 
1.5933 

1.6134 
1.6332 
1.6525 
1.6696 1.6715 
1.6883 1.6901 



0.0296 
0.1222 
0.2070 
0.2852 
0.3577 

0.4253 
0.4886 
0.5481 
0.6043 
0.6575 

0.7080 
0.7561 
0.8020 
0.8459 
0.8879 

0.9282 
0.9670 
1.0043 
1 . 0403 
1.0750 

1.1086 
1.1410 
1.1725 
1.2030 
1.2326 

1.2613 
1 . 2892 
1.3164 
1.3429 
1.3686 

1.3938 
1.4183 
1.4422 
1.4656 
1.4884 

1.5107 
1.5326 
1.5539 
1.5749 
1.5953 

1.6154 
1.6351 
1.6545 
1.6734 
1.6919 



0.0392 
0.1310 
0.2151 
0.2927 
0.3646 

0.4318 
0.4947 
0.5539 
0.6098 
0.6627 

0.7130 
0.7608 
0.8065 
0.8502 
0.8920 

0.9322 
0.9708 
1.0080 
1.0438 
1.0784 

1.1119 
1.1442 
1.1756 
1.2060 
1.2355 

1.2641 
1.2920 
1.3191 
1.3455 
1.3712 

1.3962 
1.4207 
1.4446 
1.4679 
1.4907 



5129 
5347 
55^0 
5769 



1 
1 
1 
1 
1.5974 

1.6174 
1.6371 
1.6563 
1.6753 
1.6938 



0.0488 
0.1398 
0.2231 
0.3001 
0.3716 

0.4383 
0.5008 
0.5596 
0.6152 
0.6678 

0.7178 
0.7655 
0.8109 
0.8544 
0.8961 

0.9361 
0.9746 
1.0116 
1.0473 
1.0818 

1.1151 
1.1474 
1.1787 
1.2090 
1.2384 

1.2670 
1.2947 
1.3218 
1.3481 



1.3737 1.3762 



1.3987 
1.4231 
1.4469 
1.4702 
1.4929 

1.5151 

1.5369 
1.5581 
1.5790 
1.5994 

1.6194 



1.6390 
1.6582 
1.6771 
1.6956 



0.0583 
0.1484 
0.2311 
0.3075 
0.3784 

0.4447 
0.5068 
0.5653 
0.6206 
0.6729 

0.7227 
0.7701 
0.8154 
0.8587 
0.9001 

0.9400 
0.9783 
1.0152 
1.0508 
1.0852 

1.1184 
1.1506 
1.1817 
1.2119 
1.2413 

1.2698 
1.2975 
1.3244 
1.3507 



1.4012 
1.4255 
1.4493 
1.4725 
1.4951 

1.5173 
1.5390 
1.5603 
1.5810 
1.6014 

1.6214 
1.6409 
1.6601 
1.6790 
1.6975 



0.0677 
0.1570 
0.2390 
0.3148 
0.3853 

0.4511 
0.5128 
0.5710 
0.6258 
0.6780 

0.7276 
0.7747 
0.8198 
0.8629 
0.9042 

0.9439 
0.9820 
1.0189 
1.0543 
1.0886 

1.1217 
1.1537 
1.1848 
1.2149 
1.2442 

1.2726 
1.3002 
1.3271 
1.3533 
1.3788 

1.4036 
1.4279 
1.4516 
1.4748 
1.4974 

1.5195 
1.5412 
1.5624 
1.5831 
1.6034 

1.6233 
1.6429 
1.6620 
1.6808 
1.6993 



0.0770 
0.1655 
0.2469 
0.3221 
0.3920 

0.4574 
0.5188 
0.5766 
0.6313 
0.6831 

0.7324 
0.7793 
0.8242 
0.8671 
0.9083 

0.9478 
0.9858 
1.0225 
1.0578 
1.0919 

1.1249 
1.1569 
1.1878 
1-2179 
1.2470 

1.2754 
1.3029 
1.3297 
1.3558 
1.3813 

1.4061 
1.4303 
1.4540 
1.4770 
1.4996 

1.5217 
1.5433 
1.5644 
1.5852 
1.6054 

1.6253 
1.6448 
1.6639 
1.6827 
1.7011 



0.0862 
0.1740 
0.2546 
0.3293 
0.3988 

0.4637 
0.5247 
0.5822 
0.6366 
0.6881 

0.7372 
0.7839 
0.8286 
0.8713 
0.9123 

0.9517 
0.9895 
1.0260 
1.0613 
1.0953 

1.1282 
1 . 1600 
1.1909 
1.2208 
1.2499 

1.2782 
1.3056 
1.3324 
1.3584 
1.3838 

1.4085 
1.4327 
1.4563 
1.4793 
1.5019 

1.5239 
1.5454 
1.5665 
1.5872 
1.6074 

1.6273 
1.6467 
1.6658 
1.6846 
1.7029 



TABLES 



195 



NATURAL LOGARITHMS OF NUMBERS 

{Continued) 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


5.5 


1.7048 


1.7066 


1.7084 


1.7102 


1.7120 


1.7138 


1.7156 


1.7174 


1.7192 


1.7210 


5.6 


1.7228 


1.7246 


1.7263 


1.7281 


1.7299 


1.7317 


1.7334 


1.7352 


1.7370 


1.7387 


5.7 


1.7405 


1.7422 


1.7440 


1.7457 


1.7475 


1.7491 


1.7509 


1.7527 


1.7544 


1.7561 


5.8 


1.7579 


1.7596 


1.7613 


1.7630 


1.7647 


1.7664 


1.7682 


1.7699 


1.7716 


1.7733 


5.9 


1.7750 


1.7767 


1.7783 


1.7800 


1.7817 


1.7834 


1.7851 


1.7868 


1.7884 


1.7901 


6.0 


1.7918 


1.7934 


1.7951 


1.7968 


1.7984 


1.8001 


1.8017 


1.8034 


1.8050 


1.8067 


6.1 


1.8033 


1.8099 


1.8116 


1.8132 


1.8148 


1.8165 


1.8181 


1.8197 


1.8213 


1.8229 


6.2 


1.8246 


1.8262 


1.8278 


1.8294 


1.8310 


1.8326 


1.8342 


1.8358 


1.8374 


1.8390 


6.3 


1.8406 


1.8421 


1.8437 


1.8453 


1.8469 


1.8485 


1.8500 


1.8516 


1.8532 


1.8547 


6.4 


1.8563 


1.8579 


1.8594 


1.8610 


1.8625 


1.8641 


1.8656 


1.8672 


1.8687 


1.8703 


6.5 


1.8718 


1.8733 


1.8749 


1.8764 


1.8779 


1.8795 


1.8810 


1.8825 


1.8840 


1.8856 


6.6 


1.8871 


1.8886 


1.8901 


1.8916 


1.8931 


1.8946 


1.8961 


1.8976 


1.8991 


1.9006 


6.7 


1.9021 


1.9036 


1.9051 


1.9066 


1.9081 


1.9095 


1.9110 


1.9125 


1.9140 


1.9155 


6.8 


1.9169 


1.9184 


1.9199 


1.9213 


1.9228 


1.9243 


1.9257 


1.9272 


1.9286 


1.9301 


6.9 


1.9315 


1.9330 


1.9344 


1.9359 


1.9373 


1.9387 


1.9402 


1.9416 


1.9431 


1.9445 


7.0 


1.9459 


1.9473 


1.9488 


1.9502 


1.9516 


1.9530 


1.9545 


1.9559 


1.9573 


1.9587 


7.1 


1.9601 


1.9615 


1.9629 


1.9643 


1.9657 


1.9671 


1.9685 


1.9699 


1.9713 


1.9727 


7.2 


1.9741 


1.9755 


1.9769 


1.9782 


1.9796 


1.9810 


1.9824 


1.9838 


1.9851 


1.9865 


7.3 


1.9879 


1.9892 


1.9906 


1.9920 


1.9933 


1.9947 


1.9961 


1.9974 


1.9988 


2.0001 


7.4 


2.0015 


2.0028 


2.0042 


2.0055 


2.0069 


2.0082 


2.0096 


2.0109 


2.0122 


2.0136 


7.5 


2.0149 


2.0162 


2.0176 


2.0189 


2.0202 


2.0216 


2.0229 


2.0242 


2.0255 


2.0268 


7.6 


2.0282 


2.0295 


2.0308 


2.0321 


2.0334 


2.0347 


2.0360 


2.0373 


2.0386 


2.0399 


7.7 


2.0412 


2.0425 


2.0438 


2.0451 


2.0464 


2.0477 


2.0490 


2.0503 


2.0516 


2.0528 


7.8 


2.0541 


2.0554 


2.0567 


2.0580 


2.0592 


2.0605 


2.0618 


2.0631 


2.0643 


2.0656 


7.9 


2.0669 


2.0681 


2.0694 


2.0707 


2.0719 


2.0732 


2.0744 


2.0757 


2.0769 


2.0782 


8.0 


2.0794 


2.0807 


2.0819 


2.0832 


2.0844 


2.0857 


2.0869 


2.0882 


2.0894 


2.0906 


8.1 


2.0919 


2.0931 


2.0943 


2.0956 


2.0968 


2.0980 


2.0992 


2.1005 


2.1017 


2.1029 


8.2 


2.1041 


2.1054 


2.1066 


2.1078 


2.1090 


2.1102 


2.1114 


2.1126 


2.1138 


2.1151 


8.3 


2.1163 


2.1175 


2.1187 


2.1199 


2.1211 


2.1223 


2.1235 


2.1247 


2.1259 


2.1270 


8.4 


2.1282 


2.1294 


2.1306 


2.1318 


2.1330 


2.1342 


2.1354 


2.1365 


2.1377 


2.1389 


8.5 


2.1401 


2.1412 


2.1424 


2.1436 


2.1448 


2.1459 


2.1471 


2.1483 


2.1494 


2.1506 


8.6 


2.1518 


2.1529 


2.1541 


2.1552 


2.1564 


2.1576 


2.1587 


2.1599 


2.1610 


2.1622 


8.7 


2.1633 


2.1645 


2.1656 


2.1668 


2.1679 


2.1691 


2.1702 


2.1713 


2.1725 


2.1736 


8.8 


2.1748 


2.1759 


2.1770 


2.1782 


2.1793 


2.1804 


2.1816 


2.1827 


2.1838 


2.1849 


8.9 


2.1861 


2.1872 


2.1883 


2.1894 


2.1905 


2.1917 


2.1928 


2.1939 


2.1950 


2.1961 


9.0 


2.1972 


2.1983 


2.1994 


2.2006 


2.2017 


2.2028 


2.2039 


2.2050 


2.2061 


2.2072 


9.1 


2.2083 


2.2094 


2.2105 


2.2116 


2.2127 


2.2138 


2.2149 


2.2159 


2.2170 


2.2181 


9 2 


2.2192 


2.2203 


2.2214 


2.2225 


2.2235 


2.2246 


2.2257 


2.2268 


2.2279 


2.2289 


9.3 


2.2300 


2.2311 


2.2322 


2.2332 


2.2343 


2.2354 


2.2365 


2.2375 


2.2386 


2.2397 


9.4 


2.2407 


2.2418 


2.2428 


2.2439 


2.2450 


2.2460 


2.2471 


2.2481 


2.2492 


2.2502 


9.5 


2.2513 


2.2523 


2.2534 


2.2544 


2.2555 


2.2565 


2.2576 


2.2586 


2.2597 


2.2607 


9.6 


2.2618 


2.2628 


2.2638 


2.2649 


2.2659 


2.2570 


2.2680 


2.2690 


2.2701 


2.2711 


9.7 


2.2721 


2.2732 


2.2742 


2.2752 


2.2762 


2.2773 


2.2783 


2.2793 


2.2803 


2.2814 


9.8 


2.2824 


2.2834 


2.2844 


2.2854 


2.2865 


2.2875 


2.2885 


2.2895 


2.2905 


2.2915 


9.9 


2.2925 


2.2935 


2.2946 


2.2956 2.2966 


2.2976 


2.2986 


2.2996 


2.3006 


2.3016 



196 



ENGINEERING MATHEMATICS 



NATURAL LOGARITHMS (EACH INCREASED 
BY 10) OF NUMBERS FROM 0.00 TO 0.99 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


0.0 




5.395 


6.088 


6.493 


6.781 


7.004 


7.187 


7.341 


7.474 


7.592 


0.1 


7.697 


7.793 


7.880 


7.960 


8.034 


8.103 


8.167 


8.228 


8.285 


8.339 


0.2 


8.391 


8.439 


8.486 


8.530 


8.573 


8.614 


8.653 


8.691 


8.727 


8.762 


0.3 


8.796 


8.829 


8.861 


8.891 


8.921 


8.950 


8.978 


9.006 


9.032 


9.058 


0.4 


9.084 


9.108 


9.132 


9.156 


9.179 


9.201 


9.223 


9.245 


9.266 


9.287 


0.5 


9.307 


9.327 


9.346 


9.365 


9.384 


9.402 


9.420 


9.438 


9.455 


9.472 


0.6 


9.489 


9.506 


9.522 


9.538 


9.554 


9.569 


9.584 


9.600 


9.614 


9.629 


0.7 


9.643 


9.658 


9.671 


9.685 


9.699 


9.712 


9.726 


9.739 


9.752 


9.764 


0.8 


9.777 


9.789 


9.802 


9.814 


9.826 


9.837 


9.849 


9.861 


9.872 


9.883 


0.9 


9.895 


9.906 


9.917 


9.927 


9.938 


9.949 


9.959 


9.970 


9.980 


9.990 



NATURAL LOGARITHMS OF WHOLE NUMBERS 
FROM 10 TO 209 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


1 


2.303 


2.398 


2.485 


2.565 


2.639 


2.708 


2.773 


2.833 


2.890 


2.944 


2 


2.996 


3.045 


3.091 


3.136 


3.178 


3.219 


3.258 


3.296 


3.332 


3.367 


3 


3.401 


3.434 


3.466 


3.497 


3.526 


3.555 


3.584 


3.611 


3.638 


3.664 


4 


3.689 


3.714 


3.738 


3.761 


3.784 


3.807 


3.829 


3.850 


3.871 


3.892 


5 


3.912 


3.932 


3.951 


3.970 


3.989 


4.007 


4.025 


4.043 


4.060 


4.078 


6 


4.094 


4.111 


4.127 


4.143 


4.159 


4.174 


4.190 


4.205 


4.220 


4.234 


7 


4.249 


4.263 


4.277 


4.291 


4.304 


4.318 


4.331 


4.344 


4.357 


4.369 


8 


4.382 


4.394 


4.407 


4.419 


4.431 


4.443 


4.454 


4.466 


4.477 


4.489 


9 


4.500 


4.511 


4.522 


4.533 


4.543 


4.554 


4.564 


4.575 


4.585 


4.595 


10 


4.605 


4.615 


4.625 


4.635 


4.644 


4.654 


4.663 


4.673 


4.682 


4.691 


11 


4.701 


4.710 


4.719 


4.727 


4.736 


4.745 


4.754 


4.762 


4.771 


4.779 


12 


4.788 


4.796 


4.804 


4.812 


4.820 


4.828 


4.836 


4.844 


4.852 


4.860 


13 


4.868 


4.875 


4.883 


4.890 


4.898 


4.905 


4.913 


4.920 


4.927 


4.935 


14 


4.942 


4.949 


4.956 


4.963 


4.970 


4.977 


4.984 


4.990 


4.997 


5.004 


15 


5.011 


5.017 


5.024 


5.030 


5.037 


5.043 


5.050 


5.056 


5.063 


5.069 


16 


5.075 


5.081 


5.088 


5.094 


5.100 


5.106 


5.112 


5.118 


5.124 


5.130 


17 


5.136 


5.142 


5.148 


5.153 


5.159 


5.165 


5.171 


5.176 


5.182 


5.187 


18 


5.193 


5.199 


5.204 


5.210 


5.215 


5.220 


5.226 


5.231 


5.236 


5.242 


19 


5.247 


5.252 


5.258 


5.263 


5.268 


5.273 


5.278 


5 283 


5.288 


5.293 


20 


5.298 


5.303 


5.308 


5.313 


5.318 


5.323 


5.328 


5.333 


5.338 


5.342 



TABLES 



197 



LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS 



Degrees 


sin 


COS 


tan 


cot 




0°00' 


— 00 


10.0000 


— 00 


+ 00 


90° 00' 


0° 10' 


7.4637 


9.9999 


7.4637 


2.5363 


89° 50' 


0°20' 


7.7648 


9.9999 


7.7648 


2.2352 


89° 40' 


0° 30' 


7.9408 


9.9999 


7.9409 


2.0591 


89° 30' 


0°40' 


8.0658 


9.9999 


8.0658 


1.9342 


89° 20' 


0°50' 


8.1627 


9.9999 


8.1627 


1.8373 


89° 10' 


r 00' 


8.2419 


9.9999 


8.2419. 


1.7581 


89° 00' 


1° 10' 


8.3088 


9.9999 


8.3089 


1.6911 


88° 50' 


1°20' 


8.3668 


9.9999 


8.3669 


1.6331 


88° 40' 


1°30' 


8.4179 


9.9999 


8.4181 


1.5819 


88° 30' 


1°40' 


8.4637 


9.9998 


8.4638 


1.5362 


88° 20' 


1°50' 


8.5050 


9.9998 


8.5053 


1.4947 


88° 10' 


2° 00' 


8.5428 


9.9997 


8.5431 


1.4569 


88° 00' 


2° 10' 


8.5776 


9.9997 


8.5779 


1.4221 


87° 50' 


2° 20' 


8.6097 


9.9996 


8.6101 


1.3899 


87° 40' 


2° 30' 


8.6397 


9.9996 


8.6401 


1.3599 


87° 30' 


2° 40' 


8.6677 


9.9995 


8.6682 


1.3318 


87° 20' 


2° 50' 


8.6940 


9.9995 


8.6945 


1.3055 


87° 10' 


3° 00' 


8.7188 


9.9994 


8.7194 


1.2806 


87° 00' 


3° 10' 


8.7423 


9.9993 


8.7429 


1.2571 


86° 50' 


3° 20' 


8.7645 


9.9993 


8.7652 


1.2348 


86° 40' 


3° 30' 


8.7857 


9.9992 


8.7865 


1.2135 


86° 30' 


3° 40' 


8.8059 


9.9991 


8.8067 


1 . 1933 


86° 20' 


3° 50' 


8.8251 


9.9990 


8.8261 


1.1739 


86° 10' 


4° 00' 


8.8436 


9.9989 


8.8446 


1.1554 


86° 00' 


4° 10' 


8.8613 


9.9989 


8.8624 


1.1376 


85° 50' 


4° 20' 


8.8783 


9.9988 


8.8795 


1.1205 


85° 40' 


4° 30' 


8.8946 


9.9987 


8.8960 


1.1040 


85° 30' 


4° 40' 


8.9104 


9.9986 


8.9118 


1.0882 


85° 20' 


4° 50' 


8.9256 


9.9985 


8.9272 


1.0728 


85° 10' 


5° 00' 


8.9403 


9.9983 


8.9420 


1.0580 


85° 00' 


5° 10' 


8.9545 


9.9982 


8.9563 


1.0437 


84° 50' 


5° 20' 


8.9682 


9.9981 


8.9701 


1.0299 


84° 40' 


5° 30' 


8.9816 


9.9980 


8.9836 


1.0164 


84° 30' 




cos 


sin 


cot 


tan 


Degrees 



198 



ENGINEERING MATHEMATICS 



LOGARITHMIC wSINES, COSINES, TANGENTS, 
AND COTANGENTS {Continued) 



Degrees 


sin 


COS 


tan 


cot 




5° 40' 


8.9945 


9.9979 


8.9966 


1.0034 


84° 20' 


5° 50' 


9.0070 


9.9977 


9.0093 


0.9907 


84° 10' 


6° 00' 


9.0192 


9.9976 


9.0216 


0.9784 


84° 00' 


6° 10' 


9.0311 


9.9975 


9.0336 


0.9664 


83° 50' 


6° 20' 


9.0426 


9.9973 


9.0453 


0.9547 


83° 40' 


6° 30' 


9.0539 


9.9972 


9.0567 


0.9433 


83° 30' 


6° 40' 


9.0648 


9.9971 


9.0678 


0.9322 


83° 20' 


6° 50' 


9.0755 


9.9969 


9.0786 


0.9214 


83° 10' 


7° 00' 


9.0859 


9.9968 


9.0891 


0.9109 


83° 00' 


7° 10' 


• 9.0961 


9.9966 


9.0995 


0.9005 


82° 50' 


7° 20' 


9.1060 


9.9964 


9.1096 


0.8904 


82° 40' 


7° 30' 


9.1157 


9.9963 


9.1194 


0.8806 


82° 30' 


7° 40' 


9.1252 


9.9961 


9.1291 


0.8709 


82° 20' 


7° 50' 


9.1345 


9.9959 


9.1385 


0.8615 


82° 10' 


8° 00' 


9.1436 


9.9958 


9.1478 


0.8522 


82° 00' 


8° 10' 


9.1525 


9.9956 


9.1569 


0.8431 


81° 50' 


8° 20' 


9.1612 


9.9954 


9.1658 


0.8342 


81° 40' 


8° 30' 


9 . 1697 


9.9952 


9.1745 


0.8255 


81° 30' 


8° 40' 


9.1781 


9.9950 


9.1831 


0.8169 


81° 20' 


8° 50' 


9.1863 


9.9948 


9.1915 


0.8085 


81° 10' 


9° 00' 


9 . 1943 


9.9946 


9.1997 


0.8003 


81° 00' 


9° 10' 


9.2022 


9.9944 


9.2078 


0.7922 


80° 50' 


9° 20' 


9.2100 


9.9942 


9.2158 


0.7842 


80° 40' 


9° 30' 


9.2176 


9.9940 


9.2236 


0.7764 


80° 30' 


9° 40' 


9.2251 


9.9938 


9.2313 


0.7687 


80° 20' 


9° 50' 


9.2324 


9.9936 


9.2389 


0.7611 


80° 10' 


10° 00' 


9.2397 


9.9934 


9.2463 


0.7537 


80° 00' 


10° 10' 


9.2468 


9.9931 


9.2536 


0.7464 


79° 50' 


10° 20' 


9.2538 


9.9929 


9.2609 


0.7391 


79° 40' 


10° 30' 


9.2606 


9.9927 


9.2680 


0.7320 


79° 30' 


10° 40' 


9.2674 


9.9924 


9.2750 


0.7250 


79° 20' 


10° 50' 


9.2740 


9.9922 


9.2819 


0.7181 


79° 10' 


11° 00' 


9.2806 


9.9919 


9.2887 


0.7113 


79° 00' 


11° 10' 


9.2870 


9.9917 


9.2953 


0.7047 


78° 50' 




cos 


sin 


cot 


tan 


Degrees 



TABLES 



199 



LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS (Continued) 



Degrees 


sin 


COS 


tan 


cot 




11° 20' 


9.2934 


9.9914 


9.3020 


0.6980 


78° 40' 


11° 30' 


9.2997 


9.9912 


9.3085 


0.6915 


78° 30' 


11° 40' 


9.3058 


9.9909 


9.3149 


0.6851 


78° 20' 


11° 50' 


9.3119 


9.9907 


9.3212 


0.6788 


78° 10' 


12° 00' 


9.3179 


9.9904 


9.3275 


0.6725 


78° 00' 


12° 10' 


9.3238 


9.9901 


9.3336 


0.6664 


77° 50' 


12° 20' 


9.3296 


9.9899 


9.3397 


0.6603 


77° 40' 


12° 30' 


9.3353 


9.9896 


9.3458 


0.6542 


77° 30' 


12° 40' 


9.3410 


9.9893 


9.3517 


0.6483 


77° 20' 


12° 50' 


9.3466 


9.9890 


9.3576 


0.6424 


77° 10' 


13° 00' 


9.3521' 


9.9887 


9.3634 


0.6366 


77° 00' 


13° 10' 


9.3575 


9.9884 


9.3691 


0.6309 


76° 50' 


13° 20' 


9.3629 


9.9881 


9.3748 


0.6252 


76° 40' 


13° 30' 


9.3682 


9.9878 


9.3804 


0.6196 


76° 30' 


13° 40' 


9.3734 


9.9875 


9.3859 


0.6141 


76° 20' 


13° 50' 


9.3786 


9.9872 


9.3914 


0.6086 


76° 10' 


14° 00' 


9.3837 


9.9869 


9.3968 


0.6032 


76° 00' 


14° 10' 


9.3887 


9.9866 


9.4021 


0.5979 


75° 50' 


14° 20' 


9.3937 


9.9863 


9.4074 


0.5926 


75° 40' 


14° 30' 


9.3986 


9.9859 


9.4127 


0.5873 


75° 30' 


14° 40' 


9.4035 


9.9856 


9.4178 


0.5822 


75° 20' 


14° 50' 


9.4083 


9.9853 


9.4230 


0.5770 


75° 10' 


15° 00' 


9.4130 


9.9849 


9.4281 


0.5719 


75° 00' 


15° 10' 


9.4177 


9.9846 


9.4331 


0.5669 


74° 50' 


15° 20' 


9.4223 


9.9843 


9.4381 


0.5619 


74° 40' 


15° 30' 


9.4269 


9.9839 


9.4430 


0.5570 


74° 30' 


15° 40' 


9.4314 


9.9836 


9.4479 


0.5521 


74° 20' 


15° 50' 


9.4359 


9.9832 


9.4527 


0.5473 


74° 10' 


16° 00' 


9.4403 


9.9828 


9.4575 


0.5425 


74° 00' 


16° 10' 


9 .4447 


9.9825 


9.4622 


0.5378 


73° 50' 


16° 20' 


9.4491 


9.9821 


9.4669 


0.5331 


73° 40' 


16° 30' 


9.4533 


9.9817 


9.4716 


0.5284 


73° 30' 


16° 40' 


9.4576 


9.9814 


9.4762 


0.5238 


73° 20' 


16° 50' 


9.4618 


9.9810 


9.4808 


0.5192 


73° 10' 




cos 


sin 


cot 


tan 


Degrees 



200 



ENGINEERING MATHEMATICS 



LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS {Continued) 



Degrees 


sin 


COS 


tan 


cot 




17° 00' 


9.4659 


9.9806 


9.4853 


0.5147 


73° 00' 


17° 10' 


9.4700 


9.9802 


9.4898 


0.5102 


72° 50' 


17° 20' 


9.4741 


9.9798 


9.4943 


0.5057 


72° 40' 


17° 30' 


9.4781 


9.9794 


9.4987 


0.5013 


72° 30' 


17° 40' 


9.4821 


9.9790 


9.5031 


0.4969 


72° 20' 


17° 50' 


9.4861 


9.9786 


9.5075 


0.4925 


72° 10' 


18° 00' 


9.4900 


9.9782 


9.5118 


0.4882 


72° 00' 


18° 10' 


9.4939 


9.9778 


9.5161 


0.4839 


71° 50' 


18° 20' 


9.4977 


9.9774 


9.5203 


0.4797 


71° 40' 


18° 30' 


9.5015 


9.9770 


9.5245 


0.4755 


71° 30' 


18° 40' 


9.5052 


9.9765 


9.5287 


0.4713 


71° 20' 


18° 50' 


9.5090 


9.9761 


9.5329 


0.4671 


71° 10' 


19° 00' 


9.5126 


9.9757 


9.5370 


0.4630 


71° 00' 


19° 10' 


9.5163 


9.9752 


9.5411 


0.4589 


70° 50' 


19° 20' 


9.5199 


9.9748 


9.5451 


0.4549 


70° 40' 


19° 30' 


9.5235 


9.9743 


9.5491 


0.4509 


70° 30' 


19° 40' 


9.5270 


9.9739 


9.5531 


0.4469 


70° 20' 


19° 50' 


9.5306 


9.9734 
• 


9.5571 


0.4429 


70° 10' 


20° 00' 


9.5341 


9.9730 


9.5611 


0.4389 


70° 00' 


20° 10' 


9.5375 


9.9725 


9.5650 


0.4350 


69° 50' 


20° 20' 


9.5409 


9.9721 


9.5689 


0.4311 


69° 40' 


20° 30' 


9.5443 


9.9716 


9.5727 


0.4273 


69° 30' 


20° 40' 


9.5477 


9.9711 


9.5766 


0.4234 


69° 20' 


20° 50' 


9.5510 


9.9706 


9.5804 


0.4196 


69° 10' 


21° 00' 


• 9.5543 


9.9702 


9.5842 


0.4158 


69° 00' 


21° 10' 


9.5576 


9.9697 


9.5879 


0.4121 


68° 50' 


21° 20' 


9.5609 


9.9692 


9.5917 


0.4083 


68° 40' 


21° 30' 


9.5641 


9.9687 


9.5954 


0.4046 


68° 30' 


21° 40' 


9.5673 


9.9682 


9.5991 


0.4009 


68° 20' 


21° 50' 


9.5704 


9.9677 


9.6028 


0.3972 


68° 10' 


22° 00' 


9.5736 


9.9672 


9.6064 


0.3936 


68° 00' 


22° 10' 


9.5767 


9.9667 


9.6100 


0.3900 


67° 50' 


22° 20' 


9.5798 


9.9661 


9.6136 


0.3864 


67° 40' 


22° 30' 


9.5828 


9.9656 


• 9.6172 


0.3828 


67° 30' 




cos 


sin 


cot 


tan 


Degrees 



TABLES 



201 



LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS {Continued) 



Degrees 


sin 


COS 


tan 


cot 




22° 40' 


9.5859 


9.9651 


9.6208 


0.3792 


67° 20' 


22° 50' 


9.5889 


9.9646 


9.6243 


0.3757 


67° 10' 


23° 00' 


9.5919 


9.9640 


9.6279 


0.3721 


67° 00' 


23° 10' 


9.5948 


9.9635 


9.6314 


0.3686 


66° 50' 


23° 20' 


9.5978 


9.9629 


9.6348 


0.3652 


66° 40' 


23° 30' 


9.6007 


9.9624 


9.6383 


0.3617 


66° 30' 


23° 40' 


9.6036 


9.9618 


9.6417 


0.3583 


66° 20' 


23° 50' 


9.6065 


9.9613 


9.6452 


0.3548 


66° 10' 


24° 00' 


9.6093 


9.9607 


9.6486 


0.3514 


66° 00' 


24° 10' 


9.6121 


9.9602 


9.6520 


0.3480 


65° 50' 


24° 20' 


9.6149 


9.9596 


9.6553 


0.3447 


65° 40' 


24° 30' 


9.6177 


9.9590 


9.6587 


0.3413 


65° 30' 


24° 40' 


9.6205 


9.9584 


9.6620 


0.3380 


65° 20' 


24° 50' 


9.6232 


9.9579 


9.6654 


0.3346 


65° 10' 


25° 00' 


9.6259 


9.9573 


9.6687 


0.3313 


65° 00' 


25° 10' 


9.6286 


9.9567 


9.6720 


0.3280 


64° 50' 


25° 20' 


9.6313 


9.9561 


9.6752 


0.3248 


64° 40' 


25° 30' 


9.6340 


9.9555 


9.6785 


0.3215 


64° 30' 


25° 40' 


9.6366 


9.9549 


9.6817 


0.3183 


64° 20' 


25° 50' 


9.6392 


9.9543 


9.6850 


0.3150 


64° 10' 


26° 00' 


9.6418 


9.9537 


9.6882 


0.3118 


64° 00' 


26° 10' 


9.6444 


9.9530 


9.6914 


0.3086 


63° 50' 


26° 20' 


9.6470 


9.9524 


9.6946 


0.3054 


63° 40' 


26° 30' 


9.6495 


9.9518 


9.6977 


0.3023 


63° 30' 


26° 40' 


9.6521 


9.9512 


9.7009 


0.2991 


63° 20' 


26° 50' 


9.6546 


9.9505 


9.7040 


0.2960 


63° 10' 


27° 00' 


9.6570 


9.9499 


9.7072 


0.2928 


63° 00' 


27° 10' 


9.6595 


9.9492 


9.7103 


0.2897 


62° 50' 


27° 20' 


9.6620 


9.9486 


9.7134 


0.2866 


62° 40' 


27° 30' 


9.6644 


9.9479 


9.7165 


0.2835 


62° 30' 


27° 40' 


9.6668 


9.9473 


9.7196 


0.2804 


62° 20' 


27° 50' 


9.6692 


9.9466 


9.7226 


0.2774 


62° 10' 


28° 00' 


9.6716 


9.9459 


9.7257 


0.2743 


62° 00' 


28° 10' 


9.6740 


9.9453 


9.7287 


0.2713 


61° 50' 




cos 


sin 


cot 


tan 


Degrees 



202 



ENGINEERING MATHEMATICS 



LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS {:Continued) 



Degrees 


sin 


COS 


tan 


cot 




28° 20' 


9.6763 


9.9446 


9.7317 


0.2683 


61° 40' 


28° 30' 


9.6787 


9.9439 


9.7348 


0.2652 


61° 30' 


28° 40' 


9.6810 


9.9432 


9.7378 


0.2622 


61° 20' 


28° 50' 


9.6833 


9.9425 


9.7408 


0.2592 


61° 10' 


29° 00' 


9.6856 


9.9418 


9.7438 


0.2562 


61° 00' 


29° 10' 


9.6878 


9.9411 


9.7467 


0.2533 


60° 50' 


29° 20' 


9.6901 


9.9404 


9.7497 


0.2503 


60° 40' 


29° 30' 


9.6923 


9.9397 


9.7526 


0.2474 


60° 30' 


29° 40' 


9.6946 


9.9390 


9.7556 


0.2444 


60° 20' 


29° 50' 


9.6968 


9.9383 


9.7585 


0.2415 


60° 10' 


30° 00' 


9.6990 


9.9375 


9.7614 


0.2386 


60° 00' 


30° 10' 


9.7012 


9.9368 


9.7644 


0.2356 


59° 50' 


30° 20' 


9.7033 


9.9361 


9.7673 


0.2327 


59° 40' 


30° 30' 


9.7055 


9.9353 


9.7701 


0.2299 


59° 30' 


30° 40' 


9.7076 


9.9346 


9.7730 


0.2270 


59° 20' 


30° 50' 


9.7097 


9.9338 


9.7759 


0.2241 


59° 10' 


31° 00' 


9.7118 


9.9331 


9.7788 


0.2212 


59° 00' 


31° 10' 


9.7139 


9.9323 


9.7816 


0.2184 


58° 50' 


31° 20' 


9.7160 


9.9315 


9.7845 


0.2155 


58° 40' 


31° 30' 


9.7181 


9.9308 


9.7873 


0.2127 


58° 30' 


31° 40' 


9.7201 


9.9300 


9.7902 


0.2098 


58° 20' 


31° 50' 


9.7222 


9.9292 


9.7930 


0.2070 


58° 10' 


32° 00' 


9.7242 


9.9284 


9.7958 


0.2042 


58° 00' 


32° 10' 


9.7262 


9.9276 


9.7986 


0.2014 


57° 50' 


32° 20' 


9.7282 


9.9268 


9.8014 


0.1986 


57° 40' 


32° 30' 


9.7302 


9.9260 


9.8042 


0.1958 


57° 30' 


32° 40' 


9.7322 


9.9252 


9.8070 


0.1930 


57° 20' 


32° 50' 


9.7342 


9.9244 


9.8097 


0.1903 


57° 10' 


33° 00' 


9.7361 


9.9236 


9.8125 


0.1875 


57° 00' 


33° 10' 


9.7380 


9.9228 


9.8153 


0.1847 


56° 50' 


33° 20' 


9.7400 


9.9219 


9.8180 


0.1820 


56° 40' 


33° 30' 


9.7419 


9.9211 


9.8208 


0.1792 


56° 30' 


33° 40' 


9.7438 


9.9203 


9.8235 


0.1765 


56° 20' 


33° 50' 


9.7457 


9.9194 


9.8263 


0.1737 


56° 10' 




cos 


sin 


cot 


tan 


Degrees 



TABLES 



203 



LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS (Continued) 



Degrees 


sin 


COS 


tan 


cot 




34° 00' 


9.7476 


9.9186 


9.8290 


0.1710 


56° 00' 


34° 10' 


9.7494 


9.9177 


9.8317 


0.1683 


55° 50' 


34° 20' 


9.7513 


9.9169 


9.8344 


0.1656 


55° 40' 


34° 30' 


9.7531 


9.9160 


9.8371 


0.1629 


55° 30' 


34° 40' 


9.7550 


9.9151 


9.8398 


0.1602 


55° 20' 


34° 50' 


9.7568 


9.9142 


9.8425 


0.1575 


55° 10' 


35° 00' 


9.7586 


9.9134 


9.8452 


0.1548 


55° 00' 


35° 10' 


9.7604 


9.9125 


9.8479 


0.1521 


54° 50' 


35° 20' 


9.7622 


9.9116 


9.8506 


0.1494 


54° 40' 


35° 30' 


9.7640 


9.9107 


9.8533 


0.1467 


54° 30' 


35° 40' 


9.7657 


9.9098 


9.8559 


0.1441 


54° 20' 


35° 50' 


9.7675 


9.9089 


9.8586 


0.1414 


54° 10' 


36° 00' 


9.7692 


9.9080 


9.8613 


0.1387 


54° 00' 


36° 10' 


9.7710 


9.9070 


9.8639 


0.1361 


53° 50' 


36° 20' 


9.7727 


9.9061 


9.8666 


0.1334 


53° 40' 


36° 30' 


9.7744 


9.9052 


9.8692 


0.1308 


53° 30' 


36° 40' 


9.7761 


9.9042 


9.8718 


0.1282 


53° 20' 


36° 50' 


9.7778 


9.9033 


9.8745 


0.1255 


53° 10' 


37° 00' 


9.7795 


9.9023 


9.8771 


0.1229 


53° 00' 


37° 10' 


9.7811 


9.9014 


9.8797 


0.1203 


52° 50' 


37° 20' 


9.7828 


9.9004 


9.8824 


0.1176 


52° 40' 


37° 30' 


9.7844 


9.8995 


9.8850 


0.1150 


52° 30' 


37° 40' 


9.7861 


9.8985 


9.8876 


0.1124 


52° 20' 


37° 50' 


9.7877 


9.8975 


9.8902 


0.1098 


52° 10' 


38° 00' 


9.7893 


9.8965 


9.8928 


0.1072 


52° 00' 


38° 10' 


9.7910 


9.8955 


9.8954 


0.1046 


51° 50' 


38° 20' 


9.7926 


9.8945 


9.8980 


0.1020 


51° 40' 


38° 30' 


9.7941 


9.8935 


9.9006 


0.0994 


51° 30' 


38° 40' 


9.7957 


9.8925 


9.9032 


0.0968 


51° 20' 


38° 50' 


9.7973 


9.8915 


9.9058 


0.0942 


51° 10' 


39° 00' 


9.7989 


9.8905 


9.9084 


0.0916 


51° 00' 


39° 10' 


9.8004 


9.8895 


9.9110 


0.0890 


50° 50' 


39° 20' 


9.8020 


9.8884 


9.9135 


0.0865 


50° 40' 


39° 30^ 


9.8035 


9.8874 


9.9161 


0.0839 


50° 30' 


39° 40' 


9.8050 


9.8864 


9.9187 


0.0813 


50° 20' 


39° 50' 


9.8066 


9.8853 


9.9212 


0.0788 


50° 10' 




cos 


sin 


cot 


tan 


Degrees 



204 



ENGINEERING MATHEMATICS 



LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS (Continued) 



Degrees 


sin 


COS 


tan 


cot 




40° 00' 


9.8081 


9.8843 


9.9238 


0.0762 


50° 00' 


40° 10' 


9.8096 


9.8832 


9.9264 


0.0736 


49° 50' 


40° 20' 


9.8111 


9.8821 


9.9289 


0.0711 


49° 40' 


40° 30' 


9.8125 


9.8810 


9.9315 


0.0685 


49° 30' 


40° 40' 


9.8140 


9.8800 


9.9341 


0.0659 


49° 20' 


40° 50' 


9.8155 


9.8789 


9.9366 


0.0634 


49° 10' 


41° 00' 


9.8169 


9.8778 


9.9392 


0.0608 


49° 00' 


41° 10' 


9.8184 


9.8767 


9.9417 


0.0583 


48° 50' 


41° 20' 


9.8198 


9.8756 


9.9443 


0.0557 


48° 40' 


41° 30' 


9.8213 


9.8745 


9.9468 


0.0532 


48° 30' 


41° 40' 


9.8227 


9.8733 


9.9494 


0.0506 


48° 20' 


41° 50' 


9.8241 


9.8722 


9.9519 


0.0481 


48° 10' 


42° 00' 


9.8255 


9.8711 


9.9544 


0.0456 


48° 00' 


42° 10' 


9.8269 


9.8699 


9.9570 


0.0430 


47° 50' 


42° 20' 


9.8283 


9.8688 


9.9595 


0.0405 


47° 40' 


42° 30' 


9.8297 


9.8676 


9.9621 


0.0379 


47° 30' 


42° 40' 


9.8311 


9.8665 


9.9646 


0.0354 


47° 20' 


42° 50' 


9.8324 


9.8653 


9.9671 


0.0329 


47° 10' 


43° 00' 


9.8338 


9.8641 


9.9697 


0.0303 


47° 00' 


43° 10' 


9.8351 


9.8629 


9.9722 


0.0278 


46° 50' 


43° 20' 


9.8365 


9.8618 


9.9747 


0.0253 


46° 40' 


43° 30' 


9.8378 


9.8606 


9.9772 


0.0228 


46° 30' 


43° 40' 


9.8391 


9.8594 


9.9798 


0.0202 


46° 20' 


43° 50' 


9.8405 


9.8582 


9.9823 


0.0177 


46° 10' 


44° 00' 


9.8418 


9.8569 


9.9848 


0.0152 


46° 00' 


44° 10' 


9.8431 


9.8557 


9.9874 


0.0126 


45° 50' 


44° 20' 


9.8444 


9.8545 


9.9899 


0.0101 


45° 40' 


44° 30' 


9.8457 


9.8532 


9.9924 


0.0076 


45° 30' 


44° 40' 


9.8469 


9.8520 


9.9949 


0.0051 


45° 20' 


44° 50' 


9.8482 


9.8507 


9.9975 


0.0025 


45° 10' 


45° 00' 


9.8495 


9.8495 


0.0000 


0.0000 


45° 00' 




cos 


sin 


cot 


tan 


Degrees 



TABLES 



205 



NATURAL SINES, COSINES, TANGENTS, AND 
COTANGENTS 



Degrees 


sin 


COS 


tan 


cot 




0°00' 


.0000 


1.0000 


.0000 


00 


90° 00' 


0°10' 


.0029 


1.0000 


.0029 


343.77 


89° 50' 


0°20' 


.0058 


1.0000 


.0058 


171.89 


89° 40' 


0°30' 


.0087 


1.0000 


.0087 


114.59 


89° 30' 


0°40' 


.0116 


.9999 


.0116 


85.940 


89° 20' 


0°50' 


.0145 


.9999 


.0145 


68.750 


89° 10' 


1°00' 


.0175 


.9998 


.0175 


57.290 


89° 00' 


1° 10' 


.0204 


.9998 


.0204 


49.104 


88° 50' 


1°20' 


.0233 


.9997 


.0233 


42.964 


88° 40' 


1°30' 


.0262 


.9997 


.0262 


38.188 


88° 30' 


1°40' 


.0291 


.9996 


.0291 


34.368 


88° 20' 


1°50' 


.0320 


.9995 


.0320 


31.242 


88° 10' 


2° 00' 


.0349 


.9994 


.0349 


28.636 


88° 00' 


2° 10' 


.0378 


.9993 


.0378 


26.432 


87° 50' 


2° 20' 


.0407 


.9992 


.0407 


24.542 


87° 40' 


2° 30' 


.0436 


.9990 


.0437 


22 .904 


87° 30' 


2° 40' 


.0465 


.9989 


.0466 


21.470 


87° 20' 


2° 50' 


.0494 


.9988 


.0495 


20.206 


87° 10' 


3° 00' 


.0523 


.9986 


.0524 


19.081 


87° 00' 


3° 10' 


.0552 


.9985 


.0553 


18.075 


86° 50' 


3° 20' 


.0581 


.9983 


.0582 


17.169 


86° 40' 


3° 30' 


.0610 


.9981 


.0612 


16.350 


86° 30' 


3° 40' 


.0640 


.9980 


.0641 


15.605 


86° 20' 


3° 50' 


.0669 


.9978 


.0670 


14.924 


86° 10' 


4° 00' 


.0698 


.9976 


.0699 


14.301 


86° 00' 


4° 10' 


.0727 


.9974 


.0729 


13.727 


85° 50' 


4° 20' 


.0756 


.9971 


.0758 


13.197 


85° 40' 


4° 30' 


.0785 


.9969 


.0787 


12.706 


85° 30' 


4° 40' 


.0814 


.9967 


.0816 


12.251 


85° 20' 


4° 50' 


.0843 


.9964 


.0846 


11.826 


85° 10' 


5° 00' 


.0872 


.9962 


.0875 


11.430 


85° 00' 


5° 10' 


.0901 


.9959 


.0904 


11.059 


84° 50' 


5° 20' 


.0929 


.9957 


.0934 


10.712 


84° 40' 


5° 30' 


.0958 


.9954 


.0963 


10.385 


84° 30' 




cos 


sin 


cot 


tan 


Degrees 



2o6 



ENGINEERING MATHEMATICS 



NATURAL SINES, COSINES, TANGENTS, AND 
COTANGENTS {Continued) 



Degrees 


sin 


COS 


tan 


cot 




5° 40' 


.0987 


.9951 


.0992 


10.078 


84° 20' 


5° 50' 


.1016 


.9948 


.1022 


9.7882 


84° 10' 


6° 00' 


.1045 


.9945 


.1051 


9.5144 


84° 00' 


6° 10' 


.1074 


.9942 


.1080 


9.2553 


83° 50' 


6° 20' 


.1103 


.9939 


.1110 


9.0098 


83° 40' 


6° 30' 


.1132 


.9936 


.1139 


8.7769 


83° 30' 


6° 40' 


.1161 


.9932 


.1169 


8.5555 


83° 20' 


6° 50' 


.1190 


.9929 


.1198 


8.3450 


83° 10' 


7° 00' 


.1219 


.9925 


.1228 


8.1443 


83° 00' 


7° 10' 


.1248 


.9922 


.1257 


7.9530 


82° 50' 


7° 20' 


.1276 


.9918 


.1287 


7.7704 


82° 40' 


7° 30' 


.1305 


.9914 


.1317 


7.5958 


82° 30' 


7° 40' 


.1334 


.9911 


.1346 


7.4287 


82° 20' 


7° 50' 


.1363 


.9907 


.1376 


7.2687 


82° 10' 


8° 00' 


.1392 


.9903 


.1405 


7.1154 


82° 00' 


8° 10' 


.1421 


.9899 


.1435 


6.9682 


81^50' 


8° 20' 


.1449 


.9894 


.1465 


6.8269 


81° 40' 


8° 30' 


.1478 


.9890 


.1495 


6.6912 


81° 30' 


8° 40' 


.1507 


.9886 


.1524 


6.5606 


81° 20' 


8° 50' 


.1536 


.9881 


.1554 


6.4348 


81° 10' 


9° 00' 


.1564 


.9877 


.1584 


6.3138 


81° 00' 


9° 10' 


.1593 


.9872 


.1614 


6.1970 


80° 50' 


9° 20' 


.1622 


.9868 


.1644 


6.0844 


80° 40' 


9° 30' 


.1650 


.9863 


.1673 


5.9758 


80° 30' 


9° 40' 


.1679 


.9858 


.1703 


5.8708 


80° 20' 


9° 50' 


.1708 


.9853 


.1733 


5.7694 


80° 10' 


10° 00' 


.1736 


.9848 


.1763 


5.6713 


80° 00' 


10° 10' 


.1765 


.9843 


.1793 


5.5764 


79° 50' 


10° 20' 


.1794 


.9838 


.1823 


5.4845 


79° 40' 


10° 30' 


.1822 


.9833 


.1853 


5.3955 


79° 30' 


10° 40' 


.1851 


.9827 


.1883 


5.3093 


79° 20' 


10° 50' 


.1880 


.9822 


.1914 


5.2257 


79° 10' 


11° 00' 


.1908 


.9816 


.1944 


5.1446 


79° 00' 


11° 10' 


.1937 


.9811 


.1974 


5.0658 


78° 50' 




cos 


sin 


cot 


tan 


Degrees 



TABLES 



207 



NATURAL SINES, COSINES, TANGENTS, AND 
COTANGENTS {Continued) 



Degrees 


sin 


COS 


tan 


cot 




11° 20' 


.1965 


.9805 


.2004 


4.9894 


78° 40' 


11° 30' 


.1994 


.9799 


.2035 


4.9152 


78° 30' 


11° 40' 


.2022 


.9793 


.2065 


4.8430 


78° 20' 


11° 50' 


.2051 


.9787 


.2095 


4.7729 


78° 10' 


12° 00' 


.2079 


.9781 


.2126 


4.7046 


78° 00' 


12° 10' 


.2108 


.9775 


.2156 


4.6382 


77° 50' 


12° 20' 


.2136 


.9769 


.2186 


4.5736 


77° 40' 


12° 30' 


.2164 


.9763 


.2217 


4.5107 


77° 30' 


12° 40' 


.2193 


.9757 


.2247 


4.4494 


77° 20' 


12° 50' 


.2221 


.9750 


.2278 


4.3897 


77° 10' 


13° 00' 


.2250 


.9744 


.2309 


4.3315 


77° 00' 


13° 10' 


.2278 


.9737 


.2339 


4.2747 


76° 50' 


13° 20' 


.2306 


.9730 


.2370 


4.2193 


76° 40' 


13° 30' 


.2334 


.9724 


.2401 


4.1653 


76° 30' 


13° 40' 


.2363 


.9717 


.2432 


4.1126 


76° 20' 


13° 50' 


.2391 


.9710 


.2462 


4.0611 


76° 10' 


14° 00' 


.2419 


.9703 


.2493 


4.0108 


76° 00' 


14° 10' 


.2447 


.9696 


.2524 


3.9617 


75° 50' 


14° 20' 


.2476 


.9689 


.2555 


3.9136 


75° 40' 


14° 30' 


.2504 


.9681 


.2586 


3.8667 


75° 30' 


14° 40' 


.2532 


.9674 


.2617 


3.8208 


75° 20' 


14° 50' 


.2560 


.9667 


.2648 


3.7760 


75° 10' 


15° 00' 


.2588 


.9659 


.2679 


3.7321 


75° 00' 


15° 10' 


.2616 


.9652 


.2711 


3.6891 


74° 50' 


15° 20' 


.2644 


.9644 


.2742 


3.6470 


74° 40' 


15° 30' 


.2672 


.9636 


.2773 


3.6059 


74° 30' 


15° 40' 


.2700 


.9628 


.2805 


3.5656 


74° 20' 


15° 50' 


.2728 


.9621 


.2836 


3.5261 


74° 10' 


16° 00' 


.2756 


.9613 


.2867 


3.4874 


74° 00' 


16° 10' 


.2784 


.9605 


.2899 


3.4495 


73° 50' 


16° 20' 


.2812 


.9596 


.2931 


3.4124 


73° 40' 


16° 30' 


.2840 


.9588 


,2962 


3.3759 


73° 30' 


16° 40' 


.2868 


.9580 


.2994 


3.3402 


73° 20' 


16° 50' 


.2896 


.9572 


.3026 


3.3052 


73° 10' 




cos 


sin 


cot 


tan 


Degrees 



208 



ENGINEERING MATHEMATICS 



NATURAL SINES, COSINES, TANGENTS, AND 
COTANGENTS (Continued) 



Degrees 


sin 


COS 


tan 


cot 




17° 00' 


.2924 


.9563 


.3057 


3.2709 


73° 00' 


17° 10' 


.2952 


.9555 


.3089 


3.2371 


72° 50' 


17° 20' 


.2979 


.9546 


.3121 


3.2041 


72° 40' 


17° 30' 


.3007 


.9537 


.3153 


3.1716 


72° 30' 


17° 40' 


.3035 


.9528 


.3185 


3 . 1397 


72° 20' 


17° 50' 


.3062 


.9520 


.3217 


3.1084 


72° 10' 


18° 00' 


.3090 


.9511 


.3249 


3.0777 


72° 00' 


18° 10' 


.3118 


.9502 


.3281 


3.0475 


71° 50' 


18° 20' 


.3145 


.9492 


.3314 


3.0178 


71° 40' 


18° 30' 


.3173 


.9483 


.3346 


2.9887 


71° 30' 


18° 40' 


.3201 


.9474 


.3378 


2.9600 


71° 20' 


18° 50' 


.3228 


.9465 


.3411 


2.9319 


71° 10' 


19° 00' 


.3256 


.9455 


.3443 


2.9042 


71° 00' 


19° 10' 


.3283 


.9446 


.3476 


2.8770 


70° 50' 


19° 20' 


.3311 


.9436 


.3508 


2.8502 


70° 40' 


19° 30' 


.3338 


.9426 


.3541 


2.8239 


70° 30' 


19° 40' 


.3365 


.9417 


.3574 


2.7980 


70° 20' 


19° 50' 


.3393 


.9407 


.3607 


2.7725 


70° 10' 


20° 00' 


.3420 


.9397 


.3640 


2.7475 


70° 00' 


20° 10' 


.3448 


.9387 


# .3673 


2.7228 


69° 50' 


20° 20' 


.3475 


.9377 


.3706 


2.6985 


69° 40' 


20° 30' 


.3502 


.9367 


.3739 


2.6746 


69° 30' 


20° 40' 


.3529 


.9356 


.3772 


2.6511 


69° 20' 


20° 50' 


.3557 


.9346 


.3805 


2.6279 


69° 10' 


21° 00' 


.3584 


.9336 


.3839 


2.6051 


69° 00' 


21° 10' 


.3611 


.9325 


.3872 


2.5826 


68° 50' 


21° 20' 


.3638 


.9315 


.3906 


2.5605 


68° 40' 


21° 30' 


.3665 


.9304 


.3939 


2.5386 


68° 30' 


21° 40' 


.3692 


.9293 


.3973 


2.5172 


68° 20' 


21° 50' 


.3719 


.9283 


.4006 


2.4960 


68° 10' 


22° 00' 


.3746 


.9272 


.4040 


2.4751 


68° 00' 


22° 10' 


.3773 


.9261 


.4074 


2.4545 


67° 50' 


22° 20' 


.3800 


.9250 


.4108 


2.4342 


67° 40' 


22° 30' 


.3827 


.9239 


.4142 


2.4142 


67° 30' 




cos 


sin 


cot 


tan 


Degrees 



TABLES 



209 



NATURAL SINES, COSINES, TANGENTS, AND 
COTANGENTS {Continued) 



Degrees 


sin 


COS 


tan 


cot 




22° 40' 


.3854 


.9228 


.4176 


2.3945 


67° 20' 


22° 50' 


.3881 


.9216 


.4210 


2.3750 


67° 10' 


23° 00' 


.3907 


.9205 


.4245 


2.3559 


67° 00' 


23° 10' 


.3934 


.9194 


.4279 


2.3369 


66° 50' 


23° 20' 


.3961 


.9182 


.4314 


2.3183 


66° 40' 


23° 30' 


.3987 


.9171 


.4348 


2.2998 


66° 30' 


23° 40' 


.4014 


.9159 


.4383 


2.2817 


66° 20' 


23° 50' 


.4041 


.9147 


.4417 


2.2637 


66° 10' 


24° 00' 


.4067 


.9135 


.4452 


2.2460 


66° 00' 


24° 10' 


.4094 


.9124 


.4487 


2.2286 


65° 50' 


24° 20' 


.4120 


.9112 


.4522 


2.2113 


65° 40' 


24° 30' 


.4147 


.9100 


.4557 


2.1943 


65° 30' 


24° 40' 


.4173 


.9088 


.4592 


2.1775 


65° 20' 


24° 50' 


.4200 


.9075 


.4628 


2.1609 


65° 10' 


25° 00' 


.4226 


.9063 


.4663 


2.1445 


65° 00' 


25° 10' 


.4253 


.9051 


.4699 


2 . 1283 


64° 50' 


25° 20' 


.4279 


.9038 


.4734 


2.1123 


64° 40' 


25° 30' 


.4305 


.9026 


.4770 


2.0965 


64° 30' 


25° 40' 


.4331 


.9013 


.4806 


2.0809 


64° 20' 


25° 50' 


.4358 


.9001 


.4841 


2.0655 


64° 10' 


26° 00' 


.4384 


.8988 


.4877 


2.0503 


64° 00' 


26° 10' 


.4410 


.8975 


.4913 


2.0353 


63° 50' 


26° 20' 


.4436 


.8962 


.4950 


2.0204 


63° 40' 


26° 30' 


.4462 


.8949 


.4986 


2.0057 


63° 30' 


26° 40' 


.4488 


.8936 


.5022 


1.9912 


63° 20' 


26° 50' 


.4514 


.8923 


.5059 


1.9768 


63° 10' 


27° 00' 


.4540 


.8910 


.5095 


1.9626 


63° 00' 


27° 10' 


.4566 


.8897 


.5132 


1.9486 


62° 50' 


27° 20' 


.4592 


.8884 


.5169 


1.9347 


62° 40' 


27° 30' 


.4617 


.8870 


.5206 


1.9210 


62° 30' 


27° 40' 


.4643 


.8857 


.5243 


1.9074 


62° 20' 


27° 50' 


.4669 


.8843 


.5280 


1.8940 


62° 10' 


28° 00' 


.4695 


.8829 


.5317 


1.8807 


62° 00' 


28° 10' 


.4720 


.8816 


.5354 


1.8676 


61° 50' 




cos 


sin 


cot 


tan 


Degrees 



2IO 



ENGINEERING MATHEMATICS 



NATURAL SINES, COSINES, TANGENTS, AND 
COTANGENTS {Continued) 



Degrees 


sin 


COS 


tan 


cot 




28° 20' 


.4746 


.8802 


.5392 


1.8546 


61° 40' 


28° 30' 


.4772 


.8788 


.5430 


1.8418 


61° 30' 


28° 40' 


.4797 


.8774 


.5467 


1.8291 


61° 20' 


28° 50' 


.4823 


.8760 


.5505 


1.8165 


61° 10' 


29° 00' 


.4848 


.8746 


.5543 


1.8040 


61° 00' 


29° 10'. 


.4874 


.8732 


.5581 


1.7917 


60° 50' 


29° 20' 


.4899 


.8718 


.5619 


1.7796 


60° 40' 


29° 30' 


.4924 


.8704 


.5658 


1.7675 


60° 30' 


29° 40' 


.4950 


.8689 


.5696 


1.7556 


60° 20' 


29° 50' 


.4975 


.8675 


.5735 


1.7437 


60° 10' 


30° 00' 


.5000 


.8660 


.5774 


1.7321 


60° 00' 


30° 10' 


.5025 


.8646 


.5812 


1.7205 


59° 50' 


30° 20' 


.5050 


.8631 


.5851 


1.7090 


59° 40' 


30° 30' 


.5075 


.8616 


.5890 


1.6977 


59° 30' 


30° 40' 


.5100 


.8601 


.5930 


1.6864 


59° 20' 


30° 50' 


.5125 


.8587 


.5969 


1.6753 


59° 10' 


31° 00' 


.5150 


.8572 


.6009 


1.6643 


59° 00' 


31° 10' 


.5175 


.8557 


.6048 


1.6534 


58° 50' 


31° 20' 


.5200 


.8542 


.6088 


1.6426 


58° 40' 


31° 30' 


.5225 


.8526 


.6128 


1.6319 


58° 30' 


31° 40' 


.5250 


8511 


.6168 


1.6212 


58° 20' 


31° 50' 


.5275 


.8496 


.6208 


1.6107 


58° 10' 


32° 00' 


.5299 


.8480 


.6249 


1.6003 


58° 00' 


32° 10' 


.5324 


.8465 


.6289 


1.5900 


57° 50' 


32° 20' 


.5348 


.8450 


.6330 


1.5798 


57° 40' 


32° 30' 


.5373 


.8434 


.6371 


1.5697 


57° 30' 


32° 40' 


.5398 


.8418 


.6412 


1.5597 


57° 20' 


32° 50' 


.5422 


.8403 


.6453 


1.5497 


57° 10' 


33° 00' 


.5446 


.8387 


.6494 


1.5399 


57° 00' 


33° 10' 


.5471 


.8371 


.6536 


1.5301 


56° 50' 


33° 20' 


.5495 


.8355 


.6577 


1.5204 


56° 40' 


33° 30' 


.5519 


.8339 


.6619 


1.5108 


56° 30' 


33° 40' 


.5544 


.8323 


.6661 


1.5013 


56° 20' 


33° 50' 


.5568 


.8307 


.6703 


1.4919 


56° 10' 




cos 


sin 


cot 


tan 


Degrees 



TABLES 



211 



NATURAL SINES, COSINES, TANGENTS, AND 
COTANGENTS {Continued) 



Degrees 


sin 


COS 


tan 


cot 




34° 00' 


.5592 


.8290 


.6745 


1.4826 


56° 00' 


34° 10' 


.5616 


.8274 


.6787 


1.4733 


55° 50' 


34° 20' 


.5640 


.8258 


.6830 


1.4641 


55° 40' 


34° 30' 


.5664 


.8241 


.6873 


1.4550 


55° 30' 


34° 40' 


.5688 


.8225 


.6916 


1.4460 


55° 20' 


34° 50' 


.5712 


.8208 


.6959 


1.4370 


55° 10' 


35° 00' 


.5736 


.8192 


.7002 


1.4281 


55° 00' 


35° 10' 


.5760 


.8175 


.7046 


1 .4193 


54° 50' 


35° 20' 


.5783 


.8158 


.7089 


1.4106 


54° 40' 


35° 30' 


.5807 


.8141 


.7133 


1.4019 


54° 30' 


35° 40' 


.5831 


.8124 


.7177 


1.3934 


54° 20' 


35° 50' 


.5854 


.8107 


.7221 


1.3848 


54° 10' 


36° 00' 


.5878 


.8090 


.7265 


1.3764 


54° 00' 


36° 10' 


.5901 


.8073 


.7310 


1.3680 


53° 50' 


36° 20' 


.5925 


.8056 


.7355 


1.3597 


53° 40' 


36° 30' 


.5948 


.8039 


.7400 


1.3514 


53° 30' 


36° 40' 


.5972 


.8021 


.7445 


1.3432 


53° 20' 


36° 50' 


.5995 


.8004 


.7490 


1.3351 


53° 10' 


37° 00' 


.6018 


.7986 


.7536 


1.3270 


53° 00' 


37° 10' 


.6041 


.7969 


.7581 


1.3190 


52° 50' 


37° 20' 


.6065 


.7951 


.7627 


1.3111 


52° 40' 


37° 30' 


.6088 


.7934 


.7673 


1.3032 


52° 30' 


37° 40' 


.6111 


.7916 


.7720 


1.2954 


52° 20' 


37° 50' 


.6134 


.7898 


.7766 


1.2876 


52° 10' 


38° 00' 


.6157 


.7880 


.7813 


1.2799 


52° 00' 


38° 10' 


.6180 


.7862 


.7860 


1.2723 


51° 50' 


38° 20' 


.6202 


.7844 


.7907 


1.2647 


51° 40' 


?>%'' 30' 


.6225 


.7826 


.7954 


1.2572 


51° 30' 


38° 40' 


.6248 


.7808 


.8002 


1.2497 


51° 20' 


38° 50' 


.6271 


.7790 


.8050 


1.2423 


51° 10' 


39° 00' 


.6293 


.7771 


.8098 


1.2349 


51° 00' 


39° 10' 


.6316 


.7753 


.8146 


1.2276 


50° 50' 


39° 20' 


.6338 


.7735 


.8195 


1.2203 


50° 40' 


39° 30' 


.6361 


.7716 


.8243 


1.2131 


50° 30' 




cos 


sin 


cot 


tan 


Degrees 



212 



ENGINEERING MATHEMATICS 



NATURAL SINES, COSINES, TANGENTS, AND 
COTANGENTS {Continued) 



Degrees 


sin 


COS 


tan 


cot 




39° 40' 


.6383 


.7698 


.8292 


1.2059 


50° 20' 


39° 50' 


.6406 


.7679 


.8342 


1 . 1988 


50° 10' 


40° 00' 


.6428 


.7660 


.8391 


1.1918 


50° 00' 


40° 10' 


.6450 


.7642 


.8441 


1.1847 


49° 50' 


40° 20' 


.6472 


.7623 


.8491 


1.1778 


49° 40' 


40° 30' 


.6494 


.7604 


.8541 


1.1708 


49° 30' 


40° 40' 


.6517 


.7585 


.8591 


1 . 1640 


49° 20' 


40° 50' 


.6539 


.7566 


.8642 


1.1571 


49° 10' ^ 


41° 00' 


.6561 


.7547 


.8693 


1.1504 


49° 00' 


41° 10' 


.6583 


.7528 


.8744 


1.1436 


48° 50' 


41° 20' 


.6604 


.7509 


.8796 


1.1369 


48° 40' 


41° 30' 


.6626 


.7490 


.8847 


1.1303 


48° 30' 


41° 40' 


.6648 


.7470 


.8899 


1.1237 


48° 20' 


41° 50' 


.6670 


.7451 


.8952 


1.1171 


^48° 10' 


42° 00' 


.6691 


.7431 


.9004 


1.1106 


48° 00' 


42° 10' 


.6713 


.7412 


.9057 


1.1041 


47° 50' 


42° 20' 


.6734 


.7392 


.9110 


1.0977 


47° 40' 


42° 30' 


.6756 


.7373 


.9163 


1.0913 


47° 30' 


42° 40' 


.6777 


.7353 


.9217 


1.0850 


47° 20' 


42° 50' 


.6799 


.7333 


.9271 


1.0786 


47° 10' 


43° 00' 


.6820 


.7314 


.9325 


1.0724 


47° 00' 


43° 10' 


.6841 


.7294 


.9380 


1.0661 


46° 50' 


43° 20' 


.6862 


.7274 


.9435 


1.0599 


46° 40' 


43° 30' 


.6884 


.7254 


.9490 


1.0538 


46° 30' 


43° 40' 


.6905 


.7234 


.9545 


1.0477 


46° 20' 


43° 50' 


.6926 


.7214 


.9601 


1.0416 


46° 10' 


44° 00' 


.6947 


.7193 


.9657 


1.0355 


46° 00' 


44° 10' 


.6967 


.7173 


.9713 


1 .0295 


45° 50' 


44° 20' 


.6988 


.7153 


.9770 


1.0235 


45° 40' 


44° 30' 


.7009 


.7133 


.9827 


1.0176 


45° 30' 


44° 40' 


.7030 


.7112 


.9884 


1.0117 


45° 20' 


44° 50' 


.7050 


.7092 


.9942 


1.0058 


45° 10' 


45° 00' 


.7071 


.7071 


1.0000 


1.0000 


45° 00' 




cos 


sin 


cot 


tan 


Degrees 



TABLES 



213 





HYPERBOLIC SINES AND COSINES 


n 


cosh n 


sinh n 


n 


cosh n 


sinh n 


0.00 


1.0000 


0.0000 


2.05 


3.9484 


3.8196 


0.05 


1.0013 


0.0500 


2.10 


4.1443 


4.0219 


0.10 


1.0050 


0.1002 


2.15 


4.3507 


4.2342 


0.15 


1.0112 


0.1506 


2.20 


4.5679 


4.4571 


0.20 


1.0201 


0.2013 


2.25 


4.7966 


4.6912 


0.25 


1.0314 


0.2526 


2.30 


5.0372 


4.9369 


0.30 


1.0453 


0.3045 


2.35 


5.2905 


5 . 1952 


0.35 


1.0619 


0.3572 


2.40 


5.5569 


5.4662 


0.40 


1.0811 


0.4108 


2.45 


5.8373 


5.7510 


0.45 


1 . 1030 


0.4653 


2.50 


6.1323 


6.0502 


0.50 


1.1276 


0.5211 


2.55 


6.4426 


6.3645 


0.55 


1.1551 


0.5782 


2.60 


6.7690 


6.6947 


0.60 


1 . 1855 


0.6367 


2.65 


7.1123 


7.0417 


0.65 


1.2188 


0.6967 


2.70 


7.4735 


7.4063 


0.70 


1.2552 


0.7586 


2.75 


7.8533 


7.7894 


0.75 


1.2947 


0.8223 


2.80 


8.2527 


8.1919 


0.80 


1.3374 


0.8881 


2.85 


8.6728 


8.6150 


0.85 


1.3835 


0.9561 


2.90 


9.1146 


9.0596 


0.90 


1.4331 


1.0265 


2.95 


9.5791 


9.5268 


0.95 


1.4862 


1.0995 


3.00 


10.0677 


10.0179 


1.00 


1.5431 


1.1752 


3.05 


10.5814 


10.5340 


1.05 


1.6038 


1.2539 


3.10 


11.1215 


11.0765 


1.10 


1.6685 


1.3356 


3.15 


11.6895 


11.6466 


1.15 


1.7374 


1.4208 


3.20 


12.2866 


12.2459 


1.20 


1.8107 


1.5097 


3.25 


12.9146 


12.8758 


1.25 


1.8884 


1.6019 


3.30 


13.5748 


13.5379 


1.30 


1.9709 


1.6984 


3.35 


14.2689 


14.2338 


1.35 


2.0583 


1.7991 


3.40 


14.9987 


14.9654 


1.40 


2.1509 


1.9043 


3.45 


15.7661 


15.7343 


1.45 


2.2488 


2.0143 


3.50 


16.5728 


16.5426 


1.50 


2.3524 


2.1293 


3.55 


17.4210 


17.3923 


1.55 


2.4619 


2.2496 


3.60 


18.3128 


18.2855 


1.60 


2.5775 


2.3757 


3.65 


19.2503 


19.2243 


1.65 


2.6995 


2.5075 


3.70 


20.2360 


20.2113 


1.70 


2.8283 


2.6456 


3.75 


21.2723 


21.2488 


1.75 


2.9642 


2.7904 


3.80 


22.3618 


22.3394 


1.80 


3.1075 


2.9422 


3.85 


23.5072 


23.4859 


1.85 


3.2583 


3.1013 


3.90 


24.7113 


24.6911 


1.90 


3.4177 


3.2682 


3.95 


25.9773 


25.9581 


1.95 


3.5855 


3.4432 


4.00 


27.3082 


27.2899 


2.00 


3.7622 


3.6269 










214 ENGINEERING MATHEMATICS 

Numerical Constants 

TT = 3.141 592 654 
logioir = 0.497 149 873 

= 0.318 309 886 



1 



■K 

t" = 9.869 604 401 

VV = 1.772 453 851 

e = 2.718 281828 

logio e = 0.434 294 482 

loge 10 = 2.302 585 093 

logiologioe = 9.637 784 311 

logeTT = 1.144 729 886 

loge 2 = 0.693 147 181 

logio 2 = 0.301 029 996 



INDEX 



Acceleration, 78, 79, 80. 
Admittance, electric, 151. 
Air: 

flow through apertures, 131. 

flow through pipes, 132, 133. 
Algebra, 1-9. 

Alternating currents, 147-154. 
Altitude, determination of, 167. 
Ampere, 165. 

Analytic geometry, 19-30. 
Angle between two lines, 29. 
Angular distortion, 94. 
Angular measure, 82. 
Apertures, flow through: 

air and steam, 131-132. 

water, 120-12 1. 
Apothecaries' measure, 159. 
Arc, length of, 31, 32, 36. 
Areas: 

integral formulae for, 37, 38. 

of circles, table of, 185-186. 
Arithmetical progression, 2. 
Asymptotes of hyperbola, 25. 
Atmosphere, standard, 165. 
Atomic weights, 168. 
Avoirdupois weight, 159. 

Barometer: 

reduction of readings to 0° C, 
166. 

determination of altitudes by, 
167. 
Bazin's formulae for flow of water: 

in channels, 127-128. 

over weirs, 129-130. 
Beam loadings, 98-106. 
Beams: 

cantilever, 99. 

deflection of, 97. 

flexure of, 95-97. 
Belt friction, 89. 
Bending moment of a beam, 96. 



Bernoulli's theorem, hydraulics, 

121-122. 
Binomial theorem, 2. 
Boiling points of elements, 172- 

173. 
British thermal unit, 164, 174. 

Calculus, 30-54. 
Calorie, 164. 
Cantilever beam, 99. 
Capacity: 

electric, 145-147, 150. 

measures of, 159, 163. 
Catenary, 27. 
Center of gravity: 

composite sections, 74, 75. 

formulae for, 72-74. 

standard sections, table of, iii- 
113. 
Center of gyration, 77. 
Center of percussion, 77. 
Center of pressure, 119, 120. 
Centigrade thermometer, 163. 
Centrifugal force, 83. 
Channel beam, 117. 
Channels, flow of water in, 127- 

129. 
Characteristic of a logarithm, 190. 
Chezy's formula, flow of water, 127. 
Chord of circle, 11. 
Circle: 

circumference and area of, 11. 

chord of, II. 

equations of, 21, 22. 

moment of inertia of, 116. 

sector and segment of, 11. 

tangent to, 21. 
Circles, table of circumferences 

and areas of, 185, 186. 
Circular measure, 159. 
Circumferences of circles, table of, 
185, 186. 



215 



2l6 



INDEX 



Coefl5cients of linear expansion, 

175. 176. 
Collapsing of tubes, 131. 
Columns, formulae for, 1 06-1 11. 
Combinations and permutations, 

9. 
Common logarithms: 

base of, 3. 

of numbers, table of, 192, 193. 
Complex imaginary quantities, 7. 

8. 
Composition of forces, 87, 88. 
Compression: 

strength of materials, 93. 

stress due to, 92. 
Condensers, electric, 146, 147. 
Conductance, electric, 151. 
Cone, right circular: 

center of gravity of, 114. 

frustum of, 13. 

lateral surface and volume, 12. 
Constants, numerical, 214. 
Convergent series, 5. 
Conversion factors, English and 

Sietric, 160-163. 
Ccipper wire tables, 136, 137. 
Coulomb, 145. 
Cubes and cube roots of numbers, 

table of, 187-189. 
Cubic equations, 4, 39, 40. 
Curvature, radius of, 32. 
Curve tracing, 40, 41. 
Cycloid, 25. 
Cylinder, right circular: 

lateral surface and volume, 12. 

moment of inertia of, 115. 
Cylinders, stresses in, 130. 

Deflection of beams, 97. 
De Moivre's theorem, 9. 
Density of various substances, 

169-171. 
Differential calculus: 

application to geometry and 

physics, 30-32. 
formulae, 33. 
Differential equations, 58-71. 
exact equations, 69, 70. 
first order and first degree, 59- 
62. 



Differential equations — cont. 
first order and higher degree. 

62, 63. 
linear equations, 63-69. 
second order and first degree, 
70, 71. 
Direction cosines of a line, 27, 28, 29. 
Discriminant of quadratic equa- 
tion, I. 
Distance: 

between two points, 20, 28; 29 
from point to line, 20. 
from point to plane, 29. 
Distortion, angular, stress due to, 

94. 
Diverging series, 5, 6. 
Dry measure, 159. 
Dyne, 163. 

e, base of natural logarithms, 3, 6. 

Eddy current loss, 156 

Elastic curve of a beam, 96, 97. 

Elasticity, modulus of, 92, 93, 94. 

Elastic limit, 91, 93. 

Electricity, 134-157. 

Ellipse: 

circumference and area of, 11. 

definition of, 23. 

equation of, 23. 

tangent to, 24. 
Energy: 

electric, 139. 

kinetic, 85, 86. 

of flywheel, 83, 84. 
EngUsh and metric conversion 

factors, 160-163. 
English weights and measures, 

158, 159. 
Epicycloid, 26. 
Equations: 

cubic and higher, 4, 39, 40. 

quadratic, i. 

transcendental, 4, 5. 
Equivalents: 

heat, electric, and mechanical, 
164-166. 

pressure, 165, 166 
Erg, 164. 
Euler's formula for columns, 106, 

107. 



INDEX 



217 



Exponential transformations, 9. 
Exponents, i. 

Factors of safety, table of, 93. 

Fahrenheit thermometer, 163. 

Falling bodies, 80. 

Farad, 145. 

Flexure of beams, 95-97. 

Fliegner'sequations,flowofair,i3i. 

Flow of air, 131, 132, 133. 

Flow of steam, 132, 133. 

Flow of water, 120-130. 

Fluids, flow of, 131-133. 

Flywheel, 83, 84. 

Force, 85, 86. 

Force, centrifugal, 83. 

Forces: 

composition of, 87, 88. 

resolution of, 88. 
Francis' formula, flow of water, 

129. 
Frequency, electric dynamos, 139, 

148. 
Friction, 88, 89. 
Frustum of pyramid, 12. 
Frustum of right circular cone, 13. 

g, acceleration of gravity, 80. 
Generators, electric, 139, 141. 
Geometrical progression, 2, 3. 
Geometry, 10-13. 
Gordon's formula for columns, 

107, 108. 
Grashof's formula, flow of steam, 

132. 
, Gravity, acceleration of, 80. 
Gyration, radius of, 77. 

Head of water, 119. 

Heat, electric, and mechanical 

equivalents, 164-166. 
Henry, 143. 

Higher degree equations, 4, 39, 40. 
Horsepower, i6r5. 
Hydraulics, 1 19-13 1. 
Hyperbola: 

asymptotes, 25. 

definition of, 24. 

equations of, 24. 
Hyperbolic functions, 54-58. 



Hyperbolic logarithms, 3, 194-196. 
Hyperbolic sines and cosines, 

table of, 213. 
Hypocycloid, 26, 27. 
Hysteresis loss, 155. 

I-beam, 118. 

Imaginary quantities, 7, 8. 
Impact, 86, 87. 
Impedance, electric, 149, 150. 
Impulse, 86. 
Inclined plane, 89-91. 
Indeterminate forms, i, 38, 39. 
Induced voltage, 142, 143. 
Inductance, 143-145, 149, 150. 
Inductivities, table of, 146. 
Infinite series, 5, 6, 7. 
Integral calculus: 

areas, 37, 38. 

length of curves, 36. 

volumes, 38. 
Integration, methods of: 

partial fractions, 42. 

parts, 43. 

reduction formulae, 45, 46. 

substitution, 43-45. 

table of integrals, 46-54. 

J. B. Johnson's formula for 

columns, 109, no. 
Joule, 164. 

Kinetic energy, 85, 86. 
Kutter's formula for flow of water, 
128, 129. 

Length, measures of, 158, 160. 

Length of curves, integral formu- 
lae, 36. 

Linear expansion, coefficients of, 
175, 176. 

Liquid measures, 159, 163. 

Loadings of beams, 98-106. 

Logarithmic cross-section paper, 
189-190. 

Logarithms, 3, 4. 

Logarithms of numbers, tables of, 
192-196. 

Logarithms of trigonometric func- 
tions, tables of, 197-204. 



2l8 



INDEX 



Logarithm tables, use of, 190, 191. 
Losses due to flow of water in 
pipes, 122-125. 

Maclaurin's series, 35. 
Magnetism, 154-157. 
Magnets, attractive force, 155. 
Materials, mechanics of, 91-118. 
Materials, strength of, table, 93. 
Maxima and minima, 34, 35. 
Measurement, tables, 158-167. 
Mechanics of materials, 91-118. 
Mechanics, theoretical, 72-91. 
Melting points of elements, 172, 

173. 
Metric and English conversion 

factors, 160-163. 
Modulus of elasticity, 92, 93, 94. 
Mollier chart for steam, after 184. 
Moment of inertia: 

plane areas, formulae, 75, 76. 
solids, formula, 76. 
sohds, table of, 115. 
standard sections, table of, 116- 
118. 
Moment of resistance of a beam, 

96. 
Momentum, 85. 
Motion: 

circular, 82, 83. 
of a body, 78, 79. 
Motors, electric, 139, 140. 

Naperian logarithms, 3, 194-196. 
Napier's equations, flow of steam, 

132. 
Natural logarithms: 
base of, 3. 

of numbers, tables of, 194-196. 
Natural trigonometric functions, 

205-212. 
Navigation, 17-19. 
Neutral axis of beams, 95, 96. 
Neutral surface of beams, 95. 
Newton's method, solution of 

equations by, 39, 40. 
Normal to curve, slope of, 30. 
Numerical constants, 214. 

Oblique spherical triangles, 16, 17. 



Ohm, 165. 

Ohm's law, 138. 

Open channels, flow of water in, 

127-129. 
Oscillation, radius of, 77, 78. 
Overfall- weirs, 129, 130. 

Parabola: 

arc and area of, 11. 

center of gravity of, 113. 

definition of, 22. 

equations of, 22, 23. 

tangent to, 23. 
Parallelogram, 10. 
Pendulum, 85. 
Percussion, center of, 77. 
Permeability, 154. 
Permutations and combinations, 9. 
Phase, alternating currents, 148, 

151- 
Physical and chemical constants, 

168-184. 
Pipes: 

flow of water in, 1 21-127. 

stresses in, 130. 
Plane analytic geometry, 19-27. 
Plane, equations of, 29. 
Plane trigonometry, 13-16. 
Polar coordinates, transformation 

of, 20. 
Polygon, 10. 
Poundal, 163. 

Power, electric, 139, 152, 153, 154. 
Powers of numbers, table of, 187- 

189. 
Pressure: 

barometer readings, 166. 

center of, 119, 120. 

equivalents, 165, 166. 

pipes, 130. 

water, 119. 
Prism, right, 12. 
Progression: 

arithmetical, 2. 

geometrical, 2, 3. 
Projectiles, 80-82. 
Pyramid: 

center of gravity of, 114. 

frustum of, 12. 

lateral area and volume of, 12. 



INDEX 



219 



Quadratic equations, i. 

Radian, 82. 

Radius of curvature: 

formula for, 32. 

of beams, 97. 
Radius of gyration: 

formula for, 77. 

of standard sections, 116-118. 
Radius of oscillation, 77, 78. 
Rankine's formula for columns, 

107-108. 
Ratio test for infinite series, 6. 
Reactance, electric, 149, 150. 
Reciprocals of numbers, table of, 

187-189. 
Rectangular coordinates, trans- 
formation of, 20. 
Reduction formulae, integration 

by, 45, 46. 
Regular polygon, 10. 
Reluctance, 154. 
Resistance, electric: 

copper wire tables, 136, 137. 

formulae, 134, 135, 138, 139. 
Resistivity table, electric, 134. 
Resolution of forces, 88. 
Right circular cone, 12. 
Right circular cylinder: 

lateral surface and volume, 12. 

moment of inertia of, 115. 
Right prism, 12. 
Right spherical triangles, 16. 
Right triangle: 

area, 10. 

trigonometric formulae, 13. 
Ritter's formula for columns, 108, 

109. 
Rootsof numbers, table of, 187-189. 
Rotation of a body, 79. 80, 82, S3. 

Saturated steam, tables, 177-184. 
Saturation curves, magnetic, 156, 

157. 
Seconds-pendulum, 85. 
Section modulus of standard sec- 
tions, 116-118. 
Sector of circle: 
arc and area of, 11 
center of gravity of, 112. 



Segment of circle: 

ar^ and chord of, 11. 

center of gravity of, 113. 
Segment of sphere, 13. 
Series: 

infinite, 5, 6, 7. 

Maclaurin's, 35. 

standard, 6, 7. 

Taylor's, 35. 
Shafts, torsion of, 94, 95. 
Shear, 94, 97, 98. 
Simple pendulum, 85. 
Sine wave of voltage or current, 

147, 150, 151. 
Slope of tangent to curve, 30. 
Solid analytic geometry, 27-30. 
Solids of revolution, s^. 
Solution of equations, 4, 5, 39, 40. 
Specific heats, 173, 174. 
Sphere: 

moment of inertia of, 115. 

segment of, 13. 

surface and volume of, 13. 
Spherical triangles, 16, 17. 
Spherical trigonometry, 16-19. 
Squares and square roots of num- 
bers, table of, 187-189. 
Steam chart, MoUier, after 184. 
Steam: 

flow through apertures, 132. 

flow through pipes, 133. 
Steam tables, 177-184. 
Straight line, equations of, 19, 20, 

30. 
Straight line formula for columns, 

no. 
Strength of materials, table of, 93. 
Stress, mechanical, 91, 92. 
Stresses in pipes and cylinders. 

130, 131- 
Surface, measures of, 158, 160, 161. 
Surfaces of revolution, 37, 38. 
Susceptance, electric, 151. 

Tables, mathematical, 185-214. 
Tangent: 

to circle, 21. 

to ellipse, 24. 

to hyperbola, 25. 

to parabola, 23. 



220 



INDEX 



Tangent to curve, slope of, 30. 
Taylor's series, 35. 
Temperature coefficients of elec- 
trical resistance, table of, 135. 
Tension, 92. 
Thermometers, conversion of 

scales, 163. 
Three-phase circuits, 153, 154. 
Torque of motor, 141, 142. 
Torsion of shafts, 94, 95. 
Tracing curves, 40, 41. 
Transcendental equations, 4, 5. 
Transformation of coordinates, 20. 
Transformations, exponential- 9. 
Trapezoid: 

area of, 10. 

center of gravity of, 112. 
Triangle: 

area, 10. 

center of gravity of, iii. 

solution of, 16. 
Trigonometric formulae, 13-17. 
Trigonometric functions, tables of, 

197-212. 
Trigonometry, plane, 13-16. 
Trigonometry, spherical, 16-19. 



Troy weight, 159. 
Tubes, collapsing of, 131. 

Ultimate strength, 91, 93. 
Units: 

conversion factors, 160-163. 

heat, electric, and mechanical 
equivalents, 164-166. 

weights and measures, 158, 159. 

Velocity, 78, 79- 

Volt, 165. 

Voltage, induced, 142, 143. 

Volume, measures of, 158, 161, 162. 

Volumes: 

reduction to standard condi- 
tions, 166. 

surfaces of revolution, 37, 38. 

Water, density of, 170. 

Watt, 165. 

Weight of various substances, 169- 

171. 
Weights and measures, 158, 159. 
Weirs, 129, 130. 
Work, 85, 86. 



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